We investigate the graph associated with the p-groups of maximal class and prove a new periodic pattern describing its structure. As an application, we determine the structure of a significant …
We investigate the graph associated with the p-groups of maximal class and prove a new periodic pattern describing its structure. As an application, we determine the structure of a significant subgraph of the graph associated with the 5-groups of maximal class. This underpins a conjecture of Newman.
Let p be a prime and let B be a p-block of the symmetric group S(n) on n points. Let (D, b D ) be a Sylow B-subgroup of S(n). …
Let p be a prime and let B be a p-block of the symmetric group S(n) on n points. Let (D, b D ) be a Sylow B-subgroup of S(n). We consider the fusion system [Formula: see text] and determine a precise formula for its essential rank. In addition, the p-blocks B which admit a p-local subgroup of S(n) controlling B-fusion are characterized.
Abstract The classification of p -groups of maximal class still is a wide open problem. Coclass Conjecture W proposes a way to approach such a classification: It suggests that the …
Abstract The classification of p -groups of maximal class still is a wide open problem. Coclass Conjecture W proposes a way to approach such a classification: It suggests that the coclass graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>𝒢</m:mi></m:math> ${\mathcal{G}}$ associated with the p -groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msup><m:mi>𝒢</m:mi><m:mo>∗</m:mo></m:msup></m:math> ${{\mathcal{G}}^{\ast}}$ of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>𝒢</m:mi></m:math> ${\mathcal{G}}$ associated with those p -groups of maximal class whose automorphism group orders are divisible by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>p</m:mi><m:mo>-</m:mo><m:mn>1</m:mn></m:mrow></m:math> ${p-1}$ . We describe the broad structure of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msup><m:mi>𝒢</m:mi><m:mo>∗</m:mo></m:msup></m:math> ${{\mathcal{G}}^{\ast}}$ by determining its so-called skeleton. We investigate the smallest interesting case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>p</m:mi><m:mo>=</m:mo><m:mn>7</m:mn></m:mrow></m:math> ${p=7}$ in more detail using computational tools, and propose an explicit version of Conjecture W for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msup><m:mi>𝒢</m:mi><m:mo>∗</m:mo></m:msup></m:math> ${{\mathcal{G}}^{\ast}}$ for arbitrary <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>p</m:mi><m:mo>≥</m:mo><m:mn>7</m:mn></m:mrow></m:math> ${p\geq 7}$ . Our results are the first explicit evidence in support of Conjecture W for a coclass graph of infinite width.
Traditional lectures are commonly understood to be a teacher-centered mode of instruction where the main aim is a provision of explanations by an educator to the students. Recent literature in …
Traditional lectures are commonly understood to be a teacher-centered mode of instruction where the main aim is a provision of explanations by an educator to the students. Recent literature in higher education overwhelmingly depicts this mode of instruction as inferior compared to the desired student-centered models based on active learning techniques. First, using a four-quadrant model of educational environments, we address common confusion related to a conflation of two prevalent dichotomies by focusing on two key dimensions: (1) the extent to which students are prompted to engage actively and (2) the extent to which expert explanations are provided. Second, using a case study, we describe an evolution of tertiary mathematics education, showing how traditional instruction can still play a valuable role, provided it is suitably embedded in a student-centered course design. We support our argument by analyzing the teaching practice and learning environment in a third-year abstract algebra course through the lens of Stanislas Dehaene's theoretical framework for effective teaching and learning. The framework, comprising 'four pillars of learning', is based on a state-of-the-art conception of how learning can be facilitated according to cognitive science, educational psychology and neuroscience findings. In the case study, we illustrate how, over time, the unit design and the teaching approach have evolved into a learning environment that aligns with the four pillars of learning. We conclude that traditional lectures can and do evolve to optimize learning environments and that the erection of the dichotomy 'traditional instruction versus active learning' is no longer relevant.
We investigate the graph associated with the p-groups of maximal class. Our main result is a periodicity theorem which extends recent results of du Sautoy and of Eick and Leedham-Green. …
We investigate the graph associated with the p-groups of maximal class. Our main result is a periodicity theorem which extends recent results of du Sautoy and of Eick and Leedham-Green. We also obtain further details on the shape of the graph.
In the classification of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss …
In the classification of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss two approaches to determining the nilpotent orbits of a real symmetric pair. We apply our methods to an explicit example, and thereby classify the nilpotent orbits of SL_2(R)^4 acting on the fourth tensor power of the natural 2-dimensional SL_2(R)-module. This makes it possible to classify all stationary solutions of the so-called STU-supergravity model.
Let G be a finite group of Lie type in characteristic p. Let B be a p-block of G with non-trivial Sylow B-subgroup (D, bD). We consider the Brauer category …
Let G be a finite group of Lie type in characteristic p. Let B be a p-block of G with non-trivial Sylow B-subgroup (D, bD). We consider the Brauer category ℱ ( D , b D ) ( G , B ) and determine its essential rank. In addition, we characterize those finite groups G of Lie type that admit a p-local subgroup that controls B-fusion.
We consider the finite exceptional groups of Lie type E6+1(q)=E6(q) and E6−1(q)=E62(q), both the universal versions. We classify, up to conjugacy, the maximal p-local subgroups and radical p-subgroups of G=E6ε(q) …
We consider the finite exceptional groups of Lie type E6+1(q)=E6(q) and E6−1(q)=E62(q), both the universal versions. We classify, up to conjugacy, the maximal p-local subgroups and radical p-subgroups of G=E6ε(q) for p⩾5 with p∤q and q≡εmodp, and for p=3 with 3∤q and q≡−εmod3. As an application, the essential p-rank of the Frobenius category FD(G) is determined, where D is a Sylow p-subgroup of G. Moreover, if p=3, then we show that there is a subgroup H=F4(q) of G containing D such that FD(G)=FD(H), that is, H controls 3-fusion in G.
We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ on the Hilbert space $\mathcal{H}_4 = (\mathbb{C}^2)^{\otimes 4}$. We approach the classification …
We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ on the Hilbert space $\mathcal{H}_4 = (\mathbb{C}^2)^{\otimes 4}$. We approach the classification by realising this representation as a symmetric space of maximal rank. We first describe general methods for classifying the orbits of such a space. We then apply these methods to obtain the orbits in our special case, resulting in a complete and irredundant classification of $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$-orbits on $\mathcal{H}_4$. It follows that an element of $(\mathbb{C}^2)^{\otimes 4}$ is conjugate to an element of precisely 87 classes of elements. Each of these classes either consists of one element or of a parametrised family of elements, and the elements in the same class all have equal stabiliser in $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$. We also present a complete and irredundant classification of elements and stabilisers up to the action of ${\rm Sym}_4\ltimes\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ where ${\rm Sym}_4$ permutes the four tensor factors of $(\mathbb{C}^2)^{\otimes 4}$.
Let g be a real form of a simple complex Lie algebra.Based on ideas of Ðoković and Vinberg, we describe an algorithm to compute representatives of the nilpotent orbits of …
Let g be a real form of a simple complex Lie algebra.Based on ideas of Ðoković and Vinberg, we describe an algorithm to compute representatives of the nilpotent orbits of g using the Kostant-Sekiguchi correspondence.Our algorithms are implemented for the computer algebra system GAP and, as an application, we have built a database of nilpotent orbits of all real forms of simple complex Lie algebras of rank at most 8.In addition, we consider two real forms g and g 0 of a complex simple Lie algebra g c with Cartan decompositions g D k ˚p and g 0 D k 0 ˚p0 .We describe an explicit construction of an isomorphism g !g 0 , respecting the given Cartan decompositions, which fails if and only if g and g 0 are not isomorphic.This isomorphism can be used to map the representatives of the nilpotent orbits of g to other realizations of the same algebra.
We consider the isomorphism problem for finite abelian groups and finite meta-cyclic groups. We prove that for a dense set of positive integers $n$, isomorphism testing for abelian groups of …
We consider the isomorphism problem for finite abelian groups and finite meta-cyclic groups. We prove that for a dense set of positive integers $n$, isomorphism testing for abelian groups of black-box type of order $n$ can be done in time polynomial in $\log n$. We also prove that for a dense set of orders $n$ with given prime factors, one can test isomorphism for coprime meta-cyclic groups of black-box type of order $n$ in time polynomial in $\log n$. Prior methods for these two classes of groups have running times exponential in $\log n$.
Recent studies of [Formula: see text]-groups of coclass [Formula: see text] concentrate on the coclass graph [Formula: see text]. While the detailed structure of [Formula: see text] is unknown, it …
Recent studies of [Formula: see text]-groups of coclass [Formula: see text] concentrate on the coclass graph [Formula: see text]. While the detailed structure of [Formula: see text] is unknown, it is known that its general structure is dominated by the subgraph of ‘skeleton groups’. The original definition of these groups is technical, but some modifications for special cases have been used successfully in the literature. Given their importance, in this paper we define and investigate skeleton groups more rigorously. In particular, we study their isomorphism problem, which is a crucial step towards understanding the skeleton subgraph of [Formula: see text]. During our work we identified erroneous arguments for constructing isomorphisms in a proof of the 2013 paper on [Formula: see text]. We correct these errors here by proving the required results in a more general context.
Abstract Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. Ó Catháin and Röder described a classification …
Abstract Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. Ó Catháin and Röder described a classification algorithm for CHMs of order based on relative difference sets in groups of order ; this led to the classification of all CHMs of order at most 36. On the basis of work of de Launey and Flannery, we describe a classification algorithm for CHMs of order with a prime; we prove refined structure results and provide a classification for . Our analysis shows that every CHM of order with is equivalent to a HM with one of five distinct block structures, including Williamson‐type and (transposed) Ito matrices. If , then every CHM of order is equivalent to a Williamson‐type or (transposed) Ito matrix.
Abstract M. C. Slattery [Generation of groups of square-free order, J. Symbolic Comput. 42 2007, 6, 668–677] described computational methods to enumerate, construct and identify finite groups of squarefree order. …
Abstract M. C. Slattery [Generation of groups of square-free order, J. Symbolic Comput. 42 2007, 6, 668–677] described computational methods to enumerate, construct and identify finite groups of squarefree order. We generalise Slattery’s result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.
We show that there is a dense set of group orders such that for every such order we can decide in nearly-linear time whether two multiplication tables describe isomorphic groups. …
We show that there is a dense set of group orders such that for every such order we can decide in nearly-linear time whether two multiplication tables describe isomorphic groups. This improves significantly over the general quasi-polynomial time complexity and shows that group isomorphism can be tested efficiently for almost all group orders. We also show that in nearly-linear time it can be decided whether a multiplication table describes a group; this improves over the known super-linear complexity. Our complexities are calculated for a deterministic multi-tape Turing machine model, but we give the implications to a RAM model in the promise hierarchy as well.
In the last decade, major efforts have been made to promote inquiry-based mathematics learning at the tertiary level. The Inquiry-Based Mathematics Education (IBME) movement has gained strong momentum among some …
In the last decade, major efforts have been made to promote inquiry-based mathematics learning at the tertiary level. The Inquiry-Based Mathematics Education (IBME) movement has gained strong momentum among some mathematicians, attracting substantial funding, including from some US government agencies. This resulted in the successful mobilization of regional consortia in many states, uniting over 800 mathematics education practitioners working to reform undergraduate education. Inquiry-based learning is characterized by the fundamental premise that learners should be allowed to learn 'new to them' mathematics without being taught. This progressive idea is based on the assumption that it is best to advance learners to the level of experts by engaging learners in mathematical practices similar to those of practising mathematicians: creating new definitions, conjectures and proofs - that way learners are thought to develop 'deep mathematical understanding'. However, concerted efforts to radically reform mathematics education must be systematically scrutinized in view of available evidence and theoretical advances in the learning sciences. To that end, this scoping review sought to consolidate the extant research literature from cognitive science and educational psychology, offering a critical commentary on the effectiveness of inquiry-based learning. Our analysis of research articles and books pertaining to the topic revealed that the call for a major reform by the IBME advocates is not justified. Specifically, the general claim that students would learn better (and acquire superior conceptual understanding) if they were not taught is not supported by evidence. Neither is the general claim about the merits of IBME for addressing equity issues in mathematics classrooms.
[PLEASE SEE COMMENT] We consider the isomorphism problem for finite abelian groups and finite meta-cyclic groups. We prove that for a dense set of positive integers $n$, isomorphism testing for …
[PLEASE SEE COMMENT] We consider the isomorphism problem for finite abelian groups and finite meta-cyclic groups. We prove that for a dense set of positive integers $n$, isomorphism testing for abelian groups of black-box type of order $n$ can be done in time polynomial in $\log n$. We also prove that for a dense set of orders $n$ with given prime factors, one can test isomorphism for coprime meta-cyclic groups of black-box type of order $n$ in time polynomial in $\log n$. Prior methods for these two classes of groups have running times exponential in $\log n$.
The classification of the maximal subgroups of the Monster $\mathbf{M}$ is a long-standing problem in finite group theory. According to the literature, the classification is complete apart from the question …
The classification of the maximal subgroups of the Monster $\mathbf{M}$ is a long-standing problem in finite group theory. According to the literature, the classification is complete apart from the question of whether $\mathbf{M}$ contains maximal subgroups that are almost simple with socle $\mathrm{PSL}_2(13)$. However, this conclusion relies on reported claims, with unpublished proofs, that $\mathbf{M}$ has no maximal subgroups that are almost simple with socle $\mathrm{PSL}_2(8)$, $\mathrm{PSL}_2(16)$, or $\mathrm{PSU}_3(4)$. The aim of this paper is to settle all of these questions, and thereby complete the classification of the maximal subgroups of $\mathbf{M}$. We construct two new maximal subgroups of $\mathbf{M}$, isomorphic to the automorphism groups of $\mathrm{PSL}_2(13)$ and $\mathrm{PSU}_3(4)$, and we show that $\mathbf{M}$ has no almost simple maximal subgroup with socle $\mathrm{PSL}_2(8)$ or $\mathrm{PSL}_2(16)$. In particular, we correct the claim that $\mathbf{M}$ has no almost simple maximal subgroup with socle $\mathrm{PSU}_3(4)$. Our proofs are supported by reproducible computations carried out using the publicly available Python package mmgroup for computing with $\mathbf{M}$ recently developed by M. Seysen, and we provide explicit generators for our newly discovered maximal subgroups of $\mathbf{M}$.
Seysen's Python package mmgroup provides functionality for fast computations within the sporadic simple group $\mathbb{M}$, the Monster. The aim of this work is to present an mmgroup database of maximal …
Seysen's Python package mmgroup provides functionality for fast computations within the sporadic simple group $\mathbb{M}$, the Monster. The aim of this work is to present an mmgroup database of maximal subgroups of $\mathbb{M}$: for each conjugacy class $C$ of maximal subgroups in $\mathbb{M}$, we construct explicit group elements in mmgroup and prove that these elements generate a group in $C$. Our generators and the computations verifying correctness are available in accompanying code. The maximal subgroups of $\mathbb{M}$ have been classified in a number of papers spanning several decades; our work constitutes an independent verification of these constructions. We also correct the claim that $\mathbb{M}$ has a maximal subgroup $\mathrm{PSL}_2({59})$, and hence identify a new maximal subgroup $59{:}29$.
In the classification of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss …
In the classification of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss two approaches to determining the nilpotent orbits of a real symmetric pair. We apply our methods to an explicit example, and thereby classify the nilpotent orbits of SL_2(R)^4 acting on the fourth tensor power of the natural 2-dimensional SL_2(R)-module. This makes it possible to classify all stationary solutions of the so-called STU-supergravity model.
Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan …
Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan subalgebra. An efficient computation of this Weyl group is important for the classification of regular semisimple subalgebras, real carrier algebras, and real nilpotent orbits associated with g; the latter have various applications in theoretical physics.
Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. O Cathain and Roder described a classification algorithm for …
Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. O Cathain and Roder described a classification algorithm for CHMs of order $4n$ based on relative difference sets in groups of order $8n$; this led to the classification of all CHMs of order at most 36. Based on work of de Launey and Flannery, we describe a classification algorithm for CHMs of order $4p$ with $p$ a prime; we prove refined structure results and provide a classification for $p \leqslant 13$. Our analysis shows that every CHM of order $4p$ with $p\equiv 1\bmod 4$ is equivalent to a Hadamard matrix with one of five distinct block structures, including Williamson type and (transposed) Ito matrices. If $p\equiv 3 \bmod 4$, then every CHM of order $4p$ is equivalent to a Williamson type or (transposed) Ito matrix.
For a semisimple Lie algebra over the complex numbers, Dynkin (1952) developed an algorithm to classify the regular semisimple subalgebras, up to conjugacy by the inner automorphism group. For a …
For a semisimple Lie algebra over the complex numbers, Dynkin (1952) developed an algorithm to classify the regular semisimple subalgebras, up to conjugacy by the inner automorphism group. For a graded semisimple Lie algebra over the complex numbers, Vinberg (1979) showed that a classification of a certain type of regular subalgebras (called carrier algebras) yields a classification of the nilpotent orbits in a homogeneous component of that Lie algebra. Here we consider these problems for (graded) semisimple Lie algebras over the real numbers. First, we describe an algorithm to classify the regular semisimple subalgebras of a real semisimple Lie algebra. This also yields an algorithm for listing, up to conjugacy, the carrier algebras in a real graded semisimple real algebra. We then discuss what needs to be done to obtain a classification of the nilpotent orbits from that; such classifications have applications in differential geometry and theoretical physics. Our algorithms are implemented in the language of the computer algebra system GAP, using our package CoReLG; we report on example computations.
Let g be a real form of a simple complex Lie algebra. Based on ideas of Djokovic and Vinberg, we describe an algorithm to compute representatives of the nilpotent orbits …
Let g be a real form of a simple complex Lie algebra. Based on ideas of Djokovic and Vinberg, we describe an algorithm to compute representatives of the nilpotent orbits of g using the Kostant-Sekiguchi correspondence. Our algorithms are implemented for the computer algebra system GAP and, as an application, we have built a database of nilpotent orbits of all real forms of simple complex Lie algebras of rank at most 8. In addition, we consider two real forms g and g' of a complex simple Lie algebra g^c with Cartan decompositions g= k+p and g'=k'+p'. We describe an explicit construction of an isomorphism g -> g', respecting the given Cartan decompositions, which fails if and only if g and g' are not isomorphic. This isomorphism can be used to map the representatives of the nilpotent orbits of g to other realisations of the same algebra.
We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and …
We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and description often rely on specific knowledge of the parent object and its automorphisms. In many cases, questions of reproducibility and comparison arise. Here we present a categorical framework that addresses these questions. We prove that every characteristic structure is the image of a functor equipped with a natural transformation. This shifts the local description in the parent object to a global one in the ambient category. Through constructions in representation theory, such as tensor products, we can combine characteristic structure across multiple categories. Our results are constructive, stated in the language of a constructive type theory, which facilitates implementations in theorem checkers.
Seysen's Python package mmgroup provides functionality for fast computations within the sporadic simple group $\mathbb{M}$, the Monster. The aim of this work is to present an mmgroup database of maximal …
Seysen's Python package mmgroup provides functionality for fast computations within the sporadic simple group $\mathbb{M}$, the Monster. The aim of this work is to present an mmgroup database of maximal subgroups of $\mathbb{M}$: for each conjugacy class $C$ of maximal subgroups in $\mathbb{M}$, we construct explicit group elements in mmgroup and prove that these elements generate a group in $C$. Our generators and the computations verifying correctness are available in accompanying code. The maximal subgroups of $\mathbb{M}$ have been classified in a number of papers spanning several decades; our work constitutes an independent verification of these constructions. We also correct the claim that $\mathbb{M}$ has a maximal subgroup $\mathrm{PSL}_2({59})$, and hence identify a new maximal subgroup $59{:}29$.
Centraliser algebras of monomial representations of finite groups may be constructed and studied using methods similar to those employed in the study of permutation groups. Guided by results of D. …
Centraliser algebras of monomial representations of finite groups may be constructed and studied using methods similar to those employed in the study of permutation groups. Guided by results of D. G. Higman and others, we give an explicit construction for a basis of the centraliser algebra of a monomial representation. The character table of this algebra is then constructed via character sums over double cosets. We locate the theory of group-developed and cocyclic-developed Hadamard matrices within this framework. We apply Gr\"obner bases to produce a new classification of highly symmetric complex Hadamard matrices.
We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of …
We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian. This answers a question of Kunyavskii and Ostapenko.
We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of …
We present a computational approach to determine the space of almost-inner derivations of a finite dimensional Lie algebra given by a structure constant table. We also present an example of a Lie algebra for which the quotient algebra of the almost-inner derivations modulo the inner derivations is non-abelian. This answers a question of Kunyavskii and Ostapenko.
We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of …
We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral subgroups, we give an effective classification algorithm. For non-toral elementary abelian subgroups, we focus on algebraic groups of exceptional type with a view to future applications, and in this case we provide tables explicitly describing the subgroups and their local structure. We then describe how to transfer results to the corresponding finite groups of Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming work to complete the classification of elementary abelian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroups for torsion primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in finite groups of exceptional Lie type. Such classification results are important for determining the maximal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-local subgroups and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-radical subgroups, both of which play a crucial role in modular representation theory.
Abstract In 1980, Leedham-Green and Newman introduced the invariant coclass to the theory of groups of prime-power order and they proposed five far-reaching conjectures related to it. Their work has …
Abstract In 1980, Leedham-Green and Newman introduced the invariant coclass to the theory of groups of prime-power order and they proposed five far-reaching conjectures related to it. Their work has initiated a deep and fruitful research project in group theory that is still ongoing today. We outline the main results of this celebrated article, we describe the history leading to it, and we survey some highlights of work inspired by these results.
We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of …
We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral subgroups, we give an effective classification algorithm. For non-toral elementary abelian subgroups, we focus on algebraic groups of exceptional type with a view to future applications, and in this case we provide tables explicitly describing the subgroups and their local structure. We then describe how to transfer results to the corresponding finite groups of Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming work to complete the classification of elementary abelian $p$-subgroups for torsion primes $p$ in finite groups of exceptional Lie type. Such classification results are important for determining the maximal $p$-local subgroups and $p$-radical subgroups, both of which play a crucial role in modular representation theory.
The classification of the maximal subgroups of the Monster $\mathbf{M}$ is a long-standing problem in finite group theory. According to the literature, the classification is complete apart from the question …
The classification of the maximal subgroups of the Monster $\mathbf{M}$ is a long-standing problem in finite group theory. According to the literature, the classification is complete apart from the question of whether $\mathbf{M}$ contains maximal subgroups that are almost simple with socle $\mathrm{PSL}_2(13)$. However, this conclusion relies on reported claims, with unpublished proofs, that $\mathbf{M}$ has no maximal subgroups that are almost simple with socle $\mathrm{PSL}_2(8)$, $\mathrm{PSL}_2(16)$, or $\mathrm{PSU}_3(4)$. The aim of this paper is to settle all of these questions, and thereby complete the classification of the maximal subgroups of $\mathbf{M}$. We construct two new maximal subgroups of $\mathbf{M}$, isomorphic to the automorphism groups of $\mathrm{PSL}_2(13)$ and $\mathrm{PSU}_3(4)$, and we show that $\mathbf{M}$ has no almost simple maximal subgroup with socle $\mathrm{PSL}_2(8)$ or $\mathrm{PSL}_2(16)$. In particular, we correct the claim that $\mathbf{M}$ has no almost simple maximal subgroup with socle $\mathrm{PSU}_3(4)$. Our proofs are supported by reproducible computations carried out using the publicly available Python package mmgroup for computing with $\mathbf{M}$ recently developed by M. Seysen, and we provide explicit generators for our newly discovered maximal subgroups of $\mathbf{M}$.
In 1980, Leedham-Green and Newman introduced the invariant coclass to the theory of groups of prime-power order and they proposed five far-reaching conjectures related to it. Their work has initiated …
In 1980, Leedham-Green and Newman introduced the invariant coclass to the theory of groups of prime-power order and they proposed five far-reaching conjectures related to it. Their work has initiated a deep and fruitful research project in group theory that is still ongoing today. We outline the main results of this celebrated article, we describe the history leading to it, and we survey some highlights of work inspired by these results.
The classification of the maximal subgroups of the Monster $\mathbf{M}$ is believed to be complete subject to an unpublished result of Holmes and Wilson asserting that $\mathbf{M}$ has no maximal …
The classification of the maximal subgroups of the Monster $\mathbf{M}$ is believed to be complete subject to an unpublished result of Holmes and Wilson asserting that $\mathbf{M}$ has no maximal subgroups that are almost simple with socle isomorphic to $\text{PSL}_2(8)$, $\text{PSL}_2(16)$, or $\text{PSU}_3(4)$. We prove this result for $\text{PSL}_2(16)$, with the intention that the other two cases will be dealt with in an expanded version of this paper. Our proof is supported by reproducible computations carried out using Seysen's publicly available Python package mmgroup for computing with $\mathbf{M}$.
We show that there is a dense set of group orders such that for every such order we can decide in nearly-linear time whether two multiplication tables describe isomorphic groups. …
We show that there is a dense set of group orders such that for every such order we can decide in nearly-linear time whether two multiplication tables describe isomorphic groups. This improves significantly over the general quasi-polynomial time complexity and shows that group isomorphism can be tested efficiently for almost all group orders. We also show that in nearly-linear time it can be decided whether a multiplication table describes a group; this improves over the known super-linear complexity. Our complexities are calculated for a deterministic multi-tape Turing machine model, but we give the implications to a RAM model in the promise hierarchy as well.
Abstract A linear étale representation of a complex algebraic group G is given by a complex algebraic G -module V such that G has a Zariski-open orbit in V and …
Abstract A linear étale representation of a complex algebraic group G is given by a complex algebraic G -module V such that G has a Zariski-open orbit in V and $\dim G=\dim V$ . A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions.
We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ on the Hilbert space $\mathcal{H}_4 = (\mathbb{C}^2)^{\otimes 4}$. We approach the classification …
We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ on the Hilbert space $\mathcal{H}_4 = (\mathbb{C}^2)^{\otimes 4}$. We approach the classification by realising this representation as a symmetric space of maximal rank. We first describe general methods for classifying the orbits of such a space. We then apply these methods to obtain the orbits in our special case, resulting in a complete and irredundant classification of $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$-orbits on $\mathcal{H}_4$. It follows that an element of $(\mathbb{C}^2)^{\otimes 4}$ is conjugate to an element of precisely 87 classes of elements. Each of these classes either consists of one element or of a parametrised family of elements, and the elements in the same class all have equal stabiliser in $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$. We also present a complete and irredundant classification of elements and stabilisers up to the action of ${\rm Sym}_4\ltimes\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ where ${\rm Sym}_4$ permutes the four tensor factors of $(\mathbb{C}^2)^{\otimes 4}$.
We classify states of four rebits, that is, we classify the orbits of the group $\widehat{G}(\mathbb R) = \mathrm{\mathop{SL}}(2,\mathbb R)^4$ in the space $(\mathbb R^2)^{\otimes 4}$. This is the real …
We classify states of four rebits, that is, we classify the orbits of the group $\widehat{G}(\mathbb R) = \mathrm{\mathop{SL}}(2,\mathbb R)^4$ in the space $(\mathbb R^2)^{\otimes 4}$. This is the real analogon of the well-known SLOCC operations in quantum information theory. By constructing the $\widehat{G}(\mathbb R)$-module $(\mathbb R^2)^{\otimes 4}$ via a $\mathbb Z/2\mathbb Z$-grading of the simple split real Lie algebra of type $D_4$, the orbits are divided into three groups: semisimple, nilpotent and mixed. The nilpotent orbits have been classified in Dietrich et al. (2017), yielding applications in theoretical physics (extremal black holes in the STU model of $\mathcal{N}=2, D=4$ supergravity, see Ruggeri and Trigiante (2017)). Here we focus on the semisimple and mixed orbits which we classify with recently developed methods based on Galois cohomology, see Borovoi et al. (2021). These orbits are relevant to the classification of non-extremal (or extremal over-rotating) and two-center extremal black hole solutions in the STU model.
Traditional lectures are commonly understood to be a teacher-centered mode of instruction where the main aim is a provision of explanations by an educator to the students. Recent literature in …
Traditional lectures are commonly understood to be a teacher-centered mode of instruction where the main aim is a provision of explanations by an educator to the students. Recent literature in higher education overwhelmingly depicts this mode of instruction as inferior compared to the desired student-centered models based on active learning techniques. First, using a four-quadrant model of educational environments, we address common confusion related to a conflation of two prevalent dichotomies by focusing on two key dimensions: (1) the extent to which students are prompted to engage actively and (2) the extent to which expert explanations are provided. Second, using a case study, we describe an evolution of tertiary mathematics education, showing how traditional instruction can still play a valuable role, provided it is suitably embedded in a student-centered course design. We support our argument by analyzing the teaching practice and learning environment in a third-year abstract algebra course through the lens of Stanislav Dehaene's theoretical framework for effective teaching and learning. The framework, comprising "four pillars of learning", is based on a state-of-the-art conception of how learning can be facilitated according to cognitive science, educational psychology, and neuroscience findings. In the case study, we illustrate how, over time, the unit design and the teaching approach have evolved into a learning environment that aligns with the four pillars of learning. We conclude that traditional lectures can and do evolve to optimize learning environments and that the erection of the dichotomy "traditional instruction versus active learning" is no longer relevant.
Given a finite group $G$ acting on a set $X$ let $δ_k(G,X)$ denote the proportion of elements in $G$ that have exactly $k$ fixed points in $X$. Let $\mathrm{S}_n$ denote …
Given a finite group $G$ acting on a set $X$ let $δ_k(G,X)$ denote the proportion of elements in $G$ that have exactly $k$ fixed points in $X$. Let $\mathrm{S}_n$ denote the symmetric group acting on $[n]=\{1,2,\dots,n\}$. For $A\le\mathrm{S}_m$ and $B\le\mathrm{S}_n$, the permutational wreath product $A\wr B$ has two natural actions and we give formulas for both, $δ_k(A\wr B,[m]{\times}[n])$ and $δ_k(A\wr B,[m]^{[n]})$. We prove that for $k=0$ the values of these proportions are dense in the intervals $[δ_0(B,[n]),1]$ and $[δ_0(A,[m]),1]$. Among further result, we provide estimates for $δ_0(G,[m]^{[n]})$ for subgroups $G\leq \mathrm{S}_m\wr\mathrm{S}_n$ containing $\mathrm{A}_m^{[n]}$.
In the last decade, major efforts have been made to promote inquiry-based mathematics learning at the tertiary level. The Inquiry-Based Mathematics Education (IBME) movement has gained strong momentum among some …
In the last decade, major efforts have been made to promote inquiry-based mathematics learning at the tertiary level. The Inquiry-Based Mathematics Education (IBME) movement has gained strong momentum among some mathematicians, attracting substantial funding, including from some US government agencies. This resulted in the successful mobilization of regional consortia in many states, uniting over 800 mathematics education practitioners working to reform undergraduate education. Inquiry-based learning is characterized by the fundamental premise that learners should be allowed to learn 'new to them' mathematics without being taught. This progressive idea is based on the assumption that it is best to advance learners to the level of experts by engaging learners in mathematical practices similar to those of practising mathematicians: creating new definitions, conjectures and proofs - that way learners are thought to develop 'deep mathematical understanding'. However, concerted efforts to radically reform mathematics education must be systematically scrutinized in view of available evidence and theoretical advances in the learning sciences. To that end, this scoping review sought to consolidate the extant research literature from cognitive science and educational psychology, offering a critical commentary on the effectiveness of inquiry-based learning. Our analysis of research articles and books pertaining to the topic revealed that the call for a major reform by the IBME advocates is not justified. Specifically, the general claim that students would learn better (and acquire superior conceptual understanding) if they were not taught is not supported by evidence. Neither is the general claim about the merits of IBME for addressing equity issues in mathematics classrooms.
Traditional lectures are commonly understood to be a teacher-centered mode of instruction where the main aim is a provision of explanations by an educator to the students. Recent literature in …
Traditional lectures are commonly understood to be a teacher-centered mode of instruction where the main aim is a provision of explanations by an educator to the students. Recent literature in higher education overwhelmingly depicts this mode of instruction as inferior compared to the desired student-centered models based on active learning techniques. First, using a four-quadrant model of educational environments, we address common confusion related to a conflation of two prevalent dichotomies by focusing on two key dimensions: (1) the extent to which students are prompted to engage actively and (2) the extent to which expert explanations are provided. Second, using a case study, we describe an evolution of tertiary mathematics education, showing how traditional instruction can still play a valuable role, provided it is suitably embedded in a student-centered course design. We support our argument by analyzing the teaching practice and learning environment in a third-year abstract algebra course through the lens of Stanislas Dehaene's theoretical framework for effective teaching and learning. The framework, comprising 'four pillars of learning', is based on a state-of-the-art conception of how learning can be facilitated according to cognitive science, educational psychology and neuroscience findings. In the case study, we illustrate how, over time, the unit design and the teaching approach have evolved into a learning environment that aligns with the four pillars of learning. We conclude that traditional lectures can and do evolve to optimize learning environments and that the erection of the dichotomy 'traditional instruction versus active learning' is no longer relevant.
Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. S\'enizergues and Wei\ss (ICALP2018) proved that the isomorphism problem for virtually …
Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. S\'enizergues and Wei\ss (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in $\mathsf{PSPACE}$ when the input is given in terms of so-called virtually free presentations. Here we consider the isomorphism problem for the class of \emph{plain groups}, that is, groups that are isomorphic to a free product of finitely many finite groups and finitely many copies of the infinite cyclic group. Every plain group is naturally and efficiently presented via an inverse-closed finite convergent length-reducing rewriting system. We prove that the isomorphism problem for plain groups given in this form lies in the polynomial time hierarchy, more precisely, in $\Sigma_3^{\mathsf{P}}$. This result is achieved by combining new geometric and algebraic characterisations of groups presented by inverse-closed finite convergent length-reducing rewriting systems developed in recent work of Elder and Piggott (2021) with classical finite group isomorphism results of Babai and Szemer\'edi (1984).
A linear etale representation of a complex algebraic group $G$ is given by a complex algebraic $G$-module $V$ such that $G$ has a Zariski-open orbit on $V$ and $\dim G=\dim …
A linear etale representation of a complex algebraic group $G$ is given by a complex algebraic $G$-module $V$ such that $G$ has a Zariski-open orbit on $V$ and $\dim G=\dim V$. A current line of research investigates which etale representations can occur for reductive algebraic groups. Since a complete classification seems out of reach, it is of interest to find new examples of etale representations for such groups. The aim of this note is to describe two classical constructions of Vinberg and of Bala & Carter for nilpotent orbit classifications in semisimple Lie algebras, and to determine which reductive groups and etale representations arise in these constructions. We also explain in detail the relation between these two~constructions.
The investigation of the graph $\mathcal{G}_p$ associated with the finite $p$-groups of maximal class was initiated by Blackburn (1958) and became a deep and interesting research topic since then. Leedham-Green …
The investigation of the graph $\mathcal{G}_p$ associated with the finite $p$-groups of maximal class was initiated by Blackburn (1958) and became a deep and interesting research topic since then. Leedham-Green and McKay (1976-1984) introduced skeletons of $\mathcal{G}_p$, described their importance for the structural investigation of $\mathcal{G}_p$ and exhibited their relation to algebraic number theory. Here we go one step further: we partition the skeletons into so-called Galois trees and study their general shape. In the special case $p \geq 7$ and $p \equiv 5 \bmod 6$, we show that they have a significant impact on the periodic patterns of $\mathcal{G}_p$ conjectured by Eick, Leedham-Green, Newman and O'Brien (2013). In particular, we use Galois trees to prove a conjecture by Dietrich (2010) on these periodic patterns.
A linear \'etale representation of a complex algebraic group $G$ is given by a complex algebraic $G$-module $V$ such that $G$ has a Zariski-open orbit on $V$ and $\dim G=\dim …
A linear \'etale representation of a complex algebraic group $G$ is given by a complex algebraic $G$-module $V$ such that $G$ has a Zariski-open orbit on $V$ and $\dim G=\dim V$. A current line of research investigates which \'etale representations can occur for reductive algebraic groups. Since a complete classification seems out of reach, it is of interest to find new examples of \'etale representations for such groups. The aim of this note is to describe two classical constructions of Vinberg and of Bala & Carter for nilpotent orbit classifications in semisimple Lie algebras, and to determine which reductive groups and \'etale representations arise in these constructions. We also explain in detail the relation between these two~constructions.
Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. S{\'e}nizergues and the fifth author (ICALP2018) proved that the isomorphism problem …
Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. S{\'e}nizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in $\mathsf{PSPACE}$ when the input is given in terms of so-called virtually free presentations. Here we consider the isomorphism problem for the class of \emph{plain groups}, that is, groups that are isomorphic to a free product of finitely many finite groups and finitely many copies of the infinite cyclic group. Every plain group is naturally and efficiently presented via an inverse-closed finite convergent length-reducing rewriting system. We prove that the isomorphism problem for plain groups given in this form lies in the polynomial time hierarchy, more precisely, in $\Sigma_3^{\mathsf{P}}$. This result is achieved by combining new geometric and algebraic characterisations of groups presented by inverse-closed finite convergent length-reducing rewriting systems developed in recent work of the second and third authors (2021) with classical finite group isomorphism results of Babai and Szemer\'edi (1984).
The groups whose orders factorise into at most four primes have been described (up to isomorphism) in various papers. Given such an order n, this paper exhibits a new explicit …
The groups whose orders factorise into at most four primes have been described (up to isomorphism) in various papers. Given such an order n, this paper exhibits a new explicit and compact determination of the isomorphism types of the groups of order n together with effective algorithms to enumerate, construct, and identify these groups. The algorithms are implemented for the computer algebra system GAP.
We show that there is a dense set $\ourset\subseteq \mathbb{N}$ of group orders and a constant $c$ such that for every $n\in \ourset$ we can decide in time $O(n^2(\log n)^c)$ …
We show that there is a dense set $\ourset\subseteq \mathbb{N}$ of group orders and a constant $c$ such that for every $n\in \ourset$ we can decide in time $O(n^2(\log n)^c)$ whether two $n\times n$ multiplication tables describe isomorphic groups of order $n$. This improves significantly over the general $n^{O(\log n)}$-time complexity and shows that group isomorphism can be tested efficiently for almost all group orders $n$. We also show that in time $O(n^2
(\log n)^c)$ it can be decided whether an $n\times n$ multiplication table describes a group; this improves over the known $O(n^3)$ complexity. Our complexities are calculated for a deterministic multi-tape Turing machine model. We give the implications to a RAM model in the promise hierarchy as well.
Liedtke has introduced group functors K and K˜, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to …
Liedtke has introduced group functors K and K˜, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work, we relate K and K˜ to a group functor τ arising in the construction of the non-abelian exterior square of a group. In contrast to K˜, there exist efficient algorithms for constructing τ, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when K(G,3) is a quotient of τ(G), and when τ(G) and K˜(G,3) are isomorphic.
Abstract M. C. Slattery [Generation of groups of square-free order, J. Symbolic Comput. 42 2007, 6, 668–677] described computational methods to enumerate, construct and identify finite groups of squarefree order. …
Abstract M. C. Slattery [Generation of groups of square-free order, J. Symbolic Comput. 42 2007, 6, 668–677] described computational methods to enumerate, construct and identify finite groups of squarefree order. We generalise Slattery’s result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.
Slattery (2007) described computational methods to enumerate, construct, and identify finite groups of squarefree order. We generalise Slattery's result to the class of finite groups that have cyclic Sylow subgroups …
Slattery (2007) described computational methods to enumerate, construct, and identify finite groups of squarefree order. We generalise Slattery's result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.
Slattery (2007) described computational methods to enumerate, construct, and identify finite groups of squarefree order. We generalise Slattery's result to the class of finite groups that have cyclic Sylow subgroups …
Slattery (2007) described computational methods to enumerate, construct, and identify finite groups of squarefree order. We generalise Slattery's result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.
We show that there is a dense set $\ourset\subseteq \mathbb{N}$ of group orders and a constant $c$ such that for every $n\in \ourset$ we can decide in time $O(n^2(\log n)^c)$ …
We show that there is a dense set $\ourset\subseteq \mathbb{N}$ of group orders and a constant $c$ such that for every $n\in \ourset$ we can decide in time $O(n^2(\log n)^c)$ whether two $n\times n$ multiplication tables describe isomorphic groups of order $n$. This improves significantly over the general $n^{O(\log n)}$-time complexity and shows that group isomorphism can be tested efficiently for almost all group orders $n$. We also show that in time $O(n^2 (\log n)^c)$ it can be decided whether an $n\times n$ multiplication table describes a group; this improves over the known $O(n^3)$ complexity. Our complexities are calculated for a deterministic multi-tape Turing machine model. We give the implications to a RAM model in the promise hierarchy as well.
Motivated by quotient algorithms, such as the well-known $p$-quotient or solvable quotient algorithms, we describe how to compute extensions $\tilde H$ of a finite group $H$ by a direct sum …
Motivated by quotient algorithms, such as the well-known $p$-quotient or solvable quotient algorithms, we describe how to compute extensions $\tilde H$ of a finite group $H$ by a direct sum of isomorphic simple $\mathbb{Z}_p H$-modules such that $H$ and $\tilde H$ have the same number of generators. Similar to other quotient algorithms, our description will be via a suitable covering group of $H$. Defining this covering group requires a study of the relation module, as introduced by Gaschutz in 1954. Our investigation involves so-called Fox derivatives (coming from free differential calculus) and, as a by-product, we prove that these can be naturally described via a wreath product construction. As an application, our results can be used to describe, for a given epimorphism $G\to H$ and simple $\mathbb{Z}_p H$-module $V$, the largest quotient of $G$ that maps onto $H$ with kernel isomorphic to a direct sum of copies of $V$. We also provide a description of how to compute second cohomology groups for the (not necessarily solvable) group $H$, assuming a confluent rewriting system for $H$.
Abstract Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. Ó Catháin and Röder described a classification …
Abstract Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. Ó Catháin and Röder described a classification algorithm for CHMs of order based on relative difference sets in groups of order ; this led to the classification of all CHMs of order at most 36. On the basis of work of de Launey and Flannery, we describe a classification algorithm for CHMs of order with a prime; we prove refined structure results and provide a classification for . Our analysis shows that every CHM of order with is equivalent to a HM with one of five distinct block structures, including Williamson‐type and (transposed) Ito matrices. If , then every CHM of order is equivalent to a Williamson‐type or (transposed) Ito matrix.
Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan …
Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan subalgebra. An efficient computation of this Weyl group is important for the classification of regular semisimple subalgebras, real carrier algebras, and real nilpotent orbits associated with g; the latter have various applications in theoretical physics.
Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. O Cathain and Roder described a classification algorithm for …
Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. O Cathain and Roder described a classification algorithm for CHMs of order $4n$ based on relative difference sets in groups of order $8n$; this led to the classification of all CHMs of order at most 36. Based on work of de Launey and Flannery, we describe a classification algorithm for CHMs of order $4p$ with $p$ a prime; we prove refined structure results and provide a classification for $p \leqslant 13$. Our analysis shows that every CHM of order $4p$ with $p\equiv 1\bmod 4$ is equivalent to a Hadamard matrix with one of five distinct block structures, including Williamson type and (transposed) Ito matrices. If $p\equiv 3 \bmod 4$, then every CHM of order $4p$ is equivalent to a Williamson type or (transposed) Ito matrix.
Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are …
Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $\tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $\tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $\tau(G)$, and when $\tau(G)$ and $\tilde K(G,3)$ are isomorphic.
Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan …
Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan subalgebra. An efficient computation of this Weyl group is important for the classification of regular semisimple subalgebras, real carrier algebras, and real nilpotent orbits associated with g; the latter have various applications in theoretical physics.
Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. \'O Cath\'ain and R\"oder described a classification algorithm for …
Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. \'O Cath\'ain and R\"oder described a classification algorithm for CHMs of order $4n$ based on relative difference sets in groups of order $8n$; this led to the classification of all CHMs of order at most 36. Based on work of de Launey and Flannery, we describe a classification algorithm for CHMs of order $4p$ with $p$ a prime; we prove refined structure results and provide a classification for $p \leqslant 13$. Our analysis shows that every CHM of order $4p$ with $p\equiv 1\bmod 4$ is equivalent to a Hadamard matrix with one of five distinct block structures, including Williamson type and (transposed) Ito matrices. If $p\equiv 3 \bmod 4$, then every CHM of order $4p$ is equivalent to a Williamson type or (transposed) Ito matrix.
Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are …
Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $\tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $\tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $\tau(G)$, and when $\tau(G)$ and $\tilde K(G,3)$ are isomorphic.
We show that, given a saturated fusion system, it is, under certain conditions, possible to identify SL2(q) acting on a natural module inside the normalizer of an essential subgroup. In …
We show that, given a saturated fusion system, it is, under certain conditions, possible to identify SL2(q) acting on a natural module inside the normalizer of an essential subgroup. In particular, this is the case if the fusion system is non-constrained and has only one conjugacy class of essential subgroups.
Preface to the Second Edition Preface to the First Edition List of Figures Prerequisites by Chapter Standard Notation Introduction: Closed Linear Groups Lie Algebras and Lie Groups Complex Semisimple Lie …
Preface to the Second Edition Preface to the First Edition List of Figures Prerequisites by Chapter Standard Notation Introduction: Closed Linear Groups Lie Algebras and Lie Groups Complex Semisimple Lie Algebras Universal Enveloping Algebra Compact Lie Groups Finite-Dimensional Representations Structure Theory of Semisimple Groups Advanced Structure Theory Integration Induced Representations and Branching Theorems Prehomogeneous Vector Spaces Appendices Hints for Solutions of Problems Historical Notes References Index of Notation Index
Abstract The coclass project (1980-1994) provided a new and powerful way to classify finite p-groups. This monograph gives a coherent account of the thinking from which the philosophy that lead …
Abstract The coclass project (1980-1994) provided a new and powerful way to classify finite p-groups. This monograph gives a coherent account of the thinking from which the philosophy that lead to this classification developed. The authors provide a careful summary and explanation of the many and difficult original research papers on the coclass conjecture and the structure theorem, thus elucidating the background research for those new to the area as well as for experienced researchers. The classification philosophy has lead to many new and challenging problems. Amongst those considered are problems about pro-p-groups, the Grigorchuk group, the Nottingham group as well as linear pro-p-groups. Throughout the book the authors have used a wide range of algebraic techniques and have developed from first principles, or from basic and well known results, the cohomology of groups, spectral sequences, and representation theory. This comprehensive and long-awaited survey of the recent and current research on the structure of finite p-groups will be an important reference for all group theorists.
Preface Representations, Functors and Cohomology Cohomology and Representation Theory Jon F. Carlson 1. Introduction 2. Modules over p-groups 3. Group cohomology 4. Support varieties 5. The cohomology ring of a …
Preface Representations, Functors and Cohomology Cohomology and Representation Theory Jon F. Carlson 1. Introduction 2. Modules over p-groups 3. Group cohomology 4. Support varieties 5. The cohomology ring of a dihedral group 6. Elementary abelian subgroups in cohomology and representations 7. Quillen's dimension theorem 8. Properties of support varieties 9. The rank of the group of endotrivial modules Introduction to Block Theory Radha Kessar 1. Introduction 2. Brauer pairs 3. b-Brauer pairs 4. Some structure theory 5. Alperin's weight conjecture 6. Blocks in characteristic 7. Examples of fusion systems Introduction to Fusion Systems Markus Linckelmann 1. Local structure of finite groups 2. Fusion systems 3. Normalisers and centralisers 4. Centric subgroups 5. Alperin's fusion theorem 6. Quotients of fusion systems 7. Normal fusion systems 8. Simple fusion systems 9. Normal subsystems and control of fusion Endo-permutation Modules, a Guided Tour Jacques Th'evenaz 1. Introduction 2. Endo-permutation modules 3. The Dade group 4. Examples 5. The abelian case 6. Some small groups 7. Detection of endo-trivial modules 8. Classification of endo-trivial modules 9. Detection of endo-permutation modules 10. Functorial approach 11. The dual Burnside ring 12. Rational representations and an induction theorem 13. Classification of endo-permutation modules 14. Consequences of the classification An Introduction to the Representations and Cohomology of Categories Peter Webb 1. Introduction 2. The category algebra and some preliminaries 3. Restriction and induction of representations 4. Parametrization of simple and projective representations 5. The constant functor and limits 6. Augmentation ideals, derivations and H1 7. Extensions of categories and H2 Algebraic Groups and Finite Reductive Groups An Algebraic Introduction to Complex Reflection Groups Michel Brou'e Part I. Commutative Algebra: a Crash Course 1. Notations, conventions, and prerequisites 2. Graded algebras and modules 3. Filtrations: associated graded algebras, completion 4. Finite ring extensions 5. Local or graded k-rings 6. Free resolutions and homological dimension 7. Regular sequences, Koszul complex, depth Part II. Reflection Groups 8. Reflections and roots 9. Finite group actions on regular rings 10. Ramification and reflecting pairs 11. Characterization of reflection groups 12. Generalized characteristic degrees and Steinberg theorem 13. On the co-invariant algebra 14. Isotypic components of the symmetric algebra 15. Differential operators, harmonic polynomials 16. Orlik-Solomon theorem and first applications 17. Eigenspaces Representations of Algebraic Groups Stephen Donkin 1. Algebraic groups and representations 2. Representations of semisimple groups 3. Truncation to a Levi subgroup Modular Representations of Hecke Algebras Meinolf Geck 1. Introduction 2. Harish-Chandra series and Hecke algebras 3. Unipotent blocks 4. Generic Iwahori-Hecke algebras and specializations 5. The Kazhdan-Lusztig basis and the a-function 6. Canonical basic sets and Lusztig's ring J 7. The Fock space and canonical bases 8. The theorems of Ariki and Jacon Topics in the Theory of Algebraic Groups Gary M. Seitz 1. Introduction 2. Algebraic groups: introduction 3. Morphisms of algebraic groups 4. Maximal subgroups of classical algebraic groups 5. Maximal subgroups of exceptional algebraic groups 6. On the finiteness of double coset spaces 7. Unipotent elements in classical groups 8. Unipotent classes in exceptional groups Bounds for the Orders of the Finite Subgroups of G(k) Jean-Pierre Serre Lecture I. History: Minkowski, Schur 1. Minkowski 2. Schur 3. Blichfeldt and others Lecture II. Upper Bounds 4. The invariants t and m 5. The S-bound 6. The M-bound Lecture III. Construction of large subgroups 7. Statements 8. Arithmetic methods (k = Q) 9. Proof of theorem 9 for classical groups 10. Galois twists 11. A general construction 12. Proof of theorem 9 for exceptional groups 13. Proof of theorems 10 and 11 14. The case m = 1 Index
By giving a constructive proof of Conjecture P of Newman and O'Brien, we reduce the classification of certain p-groups of coclass r to a finite calculation. For the case p=2 …
By giving a constructive proof of Conjecture P of Newman and O'Brien, we reduce the classification of certain p-groups of coclass r to a finite calculation. For the case p=2 we show that all 2-groups of coclass r can be classified by finitely many parametrised presentations. A non-constructive proof of Conjecture P was given by du Sautoy, using the theory of zeta functions. Our constructive proof uses homological algebra. It yields more precise results and detailed structure theorems for the p-groups under consideration.
In Theorem 1.2 we may put L = G, Hi = ~k_z(G) (l = 0,1 .... , k) and Hi = 1 (l >/~).We find that for 0Further, for any …
In Theorem 1.2 we may put L = G, Hi = ~k_z(G) (l = 0,1 .... , k) and Hi = 1 (l >/~).We find that for 0Further, for any group G we denote by G >~ G' >~ G" ~ ... >~ G (t-1) >1 G ~f~ >1
Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact …
Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.
We survey the problem of constructing the groups of a given finite order. We provide an extensive bibliography and outline practical algorithmic solutions to the problem. Motivated by the millennium, …
We survey the problem of constructing the groups of a given finite order. We provide an extensive bibliography and outline practical algorithmic solutions to the problem. Motivated by the millennium, we used these methods to construct the groups of order at most 2000; we report on this calculation and describe the resulting group library.
This atlas covers groups from the families of the classification of finite simple groups. Recently updated incorporating corrections
This atlas covers groups from the families of the classification of finite simple groups. Recently updated incorporating corrections
We investigate the graph associated with the p-groups of maximal class and prove a new periodic pattern describing its structure. As an application, we determine the structure of a significant …
We investigate the graph associated with the p-groups of maximal class and prove a new periodic pattern describing its structure. As an application, we determine the structure of a significant subgraph of the graph associated with the 5-groups of maximal class. This underpins a conjecture of Newman.
Abstract M. C. Slattery [Generation of groups of square-free order, J. Symbolic Comput. 42 2007, 6, 668–677] described computational methods to enumerate, construct and identify finite groups of squarefree order. …
Abstract M. C. Slattery [Generation of groups of square-free order, J. Symbolic Comput. 42 2007, 6, 668–677] described computational methods to enumerate, construct and identify finite groups of squarefree order. We generalise Slattery’s result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.
The action of the group G0 of fixed points of a semisimple automorphism θ of a reductive algebraic group G on an eigenspace V of this automorphism in the Lie …
The action of the group G0 of fixed points of a semisimple automorphism θ of a reductive algebraic group G on an eigenspace V of this automorphism in the Lie algebra g of the group G is considered. The linear groups which are obtained in this manner are called θ-groups in this paper; they have certain properties which are analogous to properties of the adjoint group. In particular, the notions of Cartan subgroup and Weyl group can be introduced for θ-groups. It is shown that the Weyl group is generated by complex reflections; from this it follows that the algebra of invariants of any θ-group is free. Bibliography: 30 titles.
For primes $p,e>2$ there are at least $p^{e-3}/e$ groups of order $p^{2e+2}$ that have equal multisets of isomorphism types of proper subgroups and proper quotient groups, isomorphic character tables, and …
For primes $p,e>2$ there are at least $p^{e-3}/e$ groups of order $p^{2e+2}$ that have equal multisets of isomorphism types of proper subgroups and proper quotient groups, isomorphic character tables, and power maps. This obstructs recent speculation concerning a path towards efficient isomorphism tests for general finite groups. These groups have a special purpose polylogarithmic-time isomorphism test.
In the classification of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss …
In the classification of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss two approaches to determining the nilpotent orbits of a real symmetric pair. We apply our methods to an explicit example, and thereby classify the nilpotent orbits of SL_2(R)^4 acting on the fourth tensor power of the natural 2-dimensional SL_2(R)-module. This makes it possible to classify all stationary solutions of the so-called STU-supergravity model.
Among the questions which have been raised concerning the structure of a connected semisimple Lie group are those relating to conjugacy of its Cartan subgroups.
Among the questions which have been raised concerning the structure of a connected semisimple Lie group are those relating to conjugacy of its Cartan subgroups.
In this paper we take a significant step forward in the classification of 3-groups of coclass 2. Several new phenomena arise. Theoretical and computational tools have been developed to deal …
In this paper we take a significant step forward in the classification of 3-groups of coclass 2. Several new phenomena arise. Theoretical and computational tools have been developed to deal with them. We identify and are able to classify an important subset of the 3-groups of coclass 2. With this classification and further extensive computations, it is possible to predict the full classification. On the basis of the work here and earlier work on the p-groups of coclass 1, we formulate another general coclass conjecture. It implies that, given a prime p and a positive integer r, a finite computation suffices to determine the p-groups of coclass r.
In most theories for the construction of finite groups with given properties a major difficulty is the ‘isomorphism problem’, which consists of specifying how one representative of each class of …
In most theories for the construction of finite groups with given properties a major difficulty is the ‘isomorphism problem’, which consists of specifying how one representative of each class of isomorphic groups may be selected from the totality of groups constructed by the process laid down. To do this we need a practical criterion for the isomorphism of two constructed groups. The main object of the present paper is to establish such a criterion in a particular case, which in spite of its simplicity is important because it gives a method for the construction of A -groups (i.e. soluble groups whose Sylow subgroups are all Abelian). All soluble groups of cube-free order are included in this class of groups, and to exemplify the application of the criterion we summarize that part of our unpublished dissertation (7) which deals in detail with the groups of order 2 2 .3 2 .5 2 .
Journal Article ON THE CLASSIFICATION OF p-GROUPS OF MAXIMAL CLASS Get access C. R. LEEDHAM-GREEN, C. R. LEEDHAM-GREEN Department of Pure Mathematics, Queen Mary CollegeLondon E1.4NS. Search for other works …
Journal Article ON THE CLASSIFICATION OF p-GROUPS OF MAXIMAL CLASS Get access C. R. LEEDHAM-GREEN, C. R. LEEDHAM-GREEN Department of Pure Mathematics, Queen Mary CollegeLondon E1.4NS. Search for other works by this author on: Oxford Academic Google Scholar SUSAN MCKAY SUSAN MCKAY Department of Pure Mathematics, Queen Mary CollegeLondon E1.4NS. Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 35, Issue 3, September 1984, Pages 293–304, https://doi.org/10.1093/qmath/35.3.293 Published: 01 September 1984 Article history Received: 04 October 1983 Published: 01 September 1984
Cartan subalgebras play an important part in the theory of Lie algebras.Our main purpose in this paper is to find all Cartan subalgebras in real semisimple Lie algebras up to …
Cartan subalgebras play an important part in the theory of Lie algebras.Our main purpose in this paper is to find all Cartan subalgebras in real semisimple Lie algebras up to conjugacy under the adjoint groups or the full automorphism groups.The problem is very simple in complex semisimple Lie algebras, because all Cartan subalgebras in a Lie algebra over an alge- braicaHy closed field of chracteristic zero are mutually conjugate.This conjugacy theorem does not hold for a Lie algebra over a field which is not algebraically closed.For example, let $\mathfrak{g}=8I(2, R)$ be the Lie algebra of a1I $2\times 2$ real matrices with trace zero, then $\mathfrak{g}$ has two Cartan subalgebras; $t\in R\}$ and $\mathfrak{h}_{2}=\{$ ( $\theta 0$ ' $\theta 0$ ) ; $\theta\in R\}$ .$\mathfrak{h}_{1}$ and $\mathfrak{h}_{2}$ are not conjugate under an inner automorphism of $\mathfrak{g}$ , because $\mathfrak{h}_{t}$ generates a non compact groupThe conjugate classes of Cartan subalgebras in a Lie algebra over a genera# field of characteristic zero were first treated by N. Iwahori and I. Satake [8].They proved the conjugacy of Cartan subalgebras for solvable Lie algebras.Later, the conjugate classes of Cartan subalgebras (or subgroups) attracted the attention of mathematicians in connection with the theory of unitary representations.I. M. Gelfand and M. I. Graev remarked that the existence of $[n/2]+1$ different conjugate classes of Cartan sub- group in $SL(n, R)$ is connected with the existence of $[n/2]+1$ different principal non degenerate series of irreducible unitary representations of $SL(n, R)$ .Harish-Chandra, interested also in this phenomenon, proved that
Abstract The admissible representations of a real reductive group G are known by work of Langlands, Knapp, Zuckerman and Vogan. This paper describes an effective algorithm for computing the irreducible …
Abstract The admissible representations of a real reductive group G are known by work of Langlands, Knapp, Zuckerman and Vogan. This paper describes an effective algorithm for computing the irreducible representations of G with regular integral infinitesimal character. The algorithm also describes structure theory of G , including the orbits of K (ℂ) (a complexified maximal compact subgroup) on the flag variety. This algorithm has been implemented on a computer by the second author, as part of the ‘Atlas of Lie Groups and Representations’ project.
Journal Article ON p-GROUPS OF MAXIMAL CLASS I Get access C. R. LEEDHAM-GREEN, C. R. LEEDHAM-GREEN Department of Pure Mathematics, Queen Mary CollegeLondon E1 4NS Search for other works by …
Journal Article ON p-GROUPS OF MAXIMAL CLASS I Get access C. R. LEEDHAM-GREEN, C. R. LEEDHAM-GREEN Department of Pure Mathematics, Queen Mary CollegeLondon E1 4NS Search for other works by this author on: Oxford Academic Google Scholar SUSAN MCKAY SUSAN MCKAY Department of Pure Mathematics, Queen Mary CollegeLondon E1 4NS Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 27, Issue 3, September 1976, Pages 297–311, https://doi.org/10.1093/qmath/27.3.297 Published: 01 September 1976 Article history Received: 15 July 1975 Published: 01 September 1976
Let g be a real form of a simple complex Lie algebra.Based on ideas of Ðoković and Vinberg, we describe an algorithm to compute representatives of the nilpotent orbits of …
Let g be a real form of a simple complex Lie algebra.Based on ideas of Ðoković and Vinberg, we describe an algorithm to compute representatives of the nilpotent orbits of g using the Kostant-Sekiguchi correspondence.Our algorithms are implemented for the computer algebra system GAP and, as an application, we have built a database of nilpotent orbits of all real forms of simple complex Lie algebras of rank at most 8.In addition, we consider two real forms g and g 0 of a complex simple Lie algebra g c with Cartan decompositions g D k ˚p and g 0 D k 0 ˚p0 .We describe an explicit construction of an isomorphism g !g 0 , respecting the given Cartan decompositions, which fails if and only if g and g 0 are not isomorphic.This isomorphism can be used to map the representatives of the nilpotent orbits of g to other realizations of the same algebra.