The British mathematician William Burnside (1852–1927) and Ferdinand Georg Frobenius (1849–1917), Professor at Zurich and Berlin universities, are considered to be the founders of the modern theory of finite groups. …
The British mathematician William Burnside (1852–1927) and Ferdinand Georg Frobenius (1849–1917), Professor at Zurich and Berlin universities, are considered to be the founders of the modern theory of finite groups. Not only did Burnside prove many important theorems, but he also laid down lines of research for the next hundred years: two Fields Medals have been awarded for work on problems suggested by him. The Theory of Groups of Finite Order, originally published in 1897, was the first major textbook on the subject. The 1911 second edition (reissued here) contains an account of Frobenius's character theory, and remained the standard reference for many years.
H is c ∗ -normal in G if there is a normal subgroup K1 ≤ G such that G = HK1 and H ∩ K1 is S-quasinormally embedded in G. …
H is c ∗ -normal in G if there is a normal subgroup K1 ≤ G such that G = HK1 and H ∩ K1 is S-quasinormally embedded in G. In this paper, we will give some characterization of supersolvability of finite groups and improve some authors’s results.
If G is a finite group, p is a prime, and x ∈ G , it is an interesting problem to place x in a convenient small (normal) subgroup of …
If G is a finite group, p is a prime, and x ∈ G , it is an interesting problem to place x in a convenient small (normal) subgroup of G, assuming some knowledge of the order of the products x y , for certain p-elements y of G.
Abstract We give an algorithm that takes as input a transitive permutation group ( G , Ω) of degree n={m\choose 2}, and decides whether or not Ω is G -isomorphic …
Abstract We give an algorithm that takes as input a transitive permutation group ( G , Ω) of degree n={m\choose 2}, and decides whether or not Ω is G -isomorphic to the action of G on the set of unordered pairs of some set Γ on which G acts 2-homogeneously. The algorithm is constructive: if a suitable action exists, then one such will be found, together with a suitable isomorphism. We give a deterministic O(sn log c n) implemention of the algorithm that assumes advance knowledge of the suborbits of ( G , Ω). This leads to deterministic O ( sn 2 ) and Monte-Carlo O ( sn log c n ) implementations that do not make this assumption.
We study cut-and-join operators for spin Hurwitz partition functions. We provide explicit expressions for these operators in terms of derivatives in $p$-variables without straightforward matrix realization, which is yet to …
We study cut-and-join operators for spin Hurwitz partition functions. We provide explicit expressions for these operators in terms of derivatives in $p$-variables without straightforward matrix realization, which is yet to be found. With the help of these expressions spin cut-and-join operators can be calculated directly and algorithmically. The reason why it is possible is the connection between mentioned operators and specially chosen Casimir operators that are easy to compute. An essential part of the connection involves shifted Q-Schur functions.
Abstract We study nilpotent groups that act faithfully on complex algebraic varieties. In the finite case, we show that when $\textbf {k}$ is a number field, a finite $p$-subgroup of …
Abstract We study nilpotent groups that act faithfully on complex algebraic varieties. In the finite case, we show that when $\textbf {k}$ is a number field, a finite $p$-subgroup of the group of polynomial automorphisms of $\textbf {k}^d$ is isomorphic to a subgroup of $\textrm {GL}_d(\textbf {k})$. In the case of infinite nilpotent group actions, we show that a finitely generated nilpotent group $H$ acting on a complex quasiprojective variety $X$ of dimension $d$ can be embedded in a $p$-adic Lie group that acts faithfully and analytically on $\textbf {Z}_p^d$. As a consequence, we show that the virtual derived length of $H$ is at most the dimension of $X$.
Let @ be a finite group, and let x be the character of an absolutely irreducible representation of @.An algebraic number field K is defined to be a split- ring …
Let @ be a finite group, and let x be the character of an absolutely irreducible representation of @.An algebraic number field K is defined to be a split- ring field of x if can be written in the field K(x), where K(x) is the field gen- erated by K and the values of x.The existence of splitting fields with suit- able properties has been of fundamental importance in the study of group characters.In [1, Lemma 1], Brauer proved the following result.(A) Let @ be a finite group of order g pare, where p is a rational prime and (p, m) 1.Then there exists an algebraic number field K with the following properties:
We consider the function r(G) = |G|-k(G), where the group G has exactly k(G) conjugacy classes.We find all G where r(G) is small and pose a number of relevant questions.
We consider the function r(G) = |G|-k(G), where the group G has exactly k(G) conjugacy classes.We find all G where r(G) is small and pose a number of relevant questions.
Any element of a finite group that generates the same cyclic subgroup as some commutator, is itself a commutator. The paper presents a completely elementary proof of this fact, whereas …
Any element of a finite group that generates the same cyclic subgroup as some commutator, is itself a commutator. The paper presents a completely elementary proof of this fact, whereas previous proofs depend on character theory.
By a Calabi-Yau threefold we mean a minimal complex projective threefold X such that Ox(Kx) -Ox and ^(Ox) = 0.This paper consists of four parts.In Section 1 we formulate an …
By a Calabi-Yau threefold we mean a minimal complex projective threefold X such that Ox(Kx) -Ox and ^(Ox) = 0.This paper consists of four parts.In Section 1 we formulate an equivariant version of Torelli Theorem of K3 surfaces with finite group action and deduce some more geometrical consequences.In Section 2, we classify Calabi-Yau threefolds with infinite fundamental group by means of their minimal splitting coverings introduced by Beauville, and deduce as its Corollary that the nef cone is a rational simplicial cone and any rational nef divisor is semi-ample provided that C2(X) = 0 on Pic(A")]R.We also derive a sufficient condition for iri(X) to be finite in terms of the Picard number in an optimal form.In Section 3, we give a fairly concrete structure Theorem concerning C2-contractions of Calabi-Yau threefolds as a generalisation and also a correction of our earlier works for simply connected ones.In Section 4, applying the results in these three sections together with Kawamata's finiteness result of the relatively minimal models of a Calabi-Yau fiber space, we show the finiteness of the isomorphism classes of C2-contractions of each Calabi-Yau threefold.As a special case, we find the finiteness of abelian pencil structures on each X up to Aut(X). Introduction.In the light of the minimal model theory, we define a Calabi-Yau threefold to be a Q-factorial terminal projective threefold X defined over C such that Ox(Kx) -Ox and /i 1 (Ox) = 0, and regard the second Chern class C2(X) as a linear form on 'Pic(X)^ through the intersection pairing, where C2(X) for a singular X is defined as C2(X) := i'*(c2(X)) via a resolution u : X -> X and is known to be well-defined (see for example [Ogl, Lemma (1.4)]).However, as is pointed out by several authors, this preferable definition of Calabi-Yau threefold has an inevitable defect: Those Calabi-Yau threefolds, such as Igusa's example ([Ig, Page 678], [Ue, Example 16.16]), that are given as an etale quotient of an abelian threefold are then included in our category.We call them of Type A. Indeed, their pathological nature sometimes prevents us from studying Calabi-Yau threefolds uniformly.For instance,(1) there are no rational curves on Calabi-Yau threefolds of Type A, while it is expected, and has been already checked in some extent, that most of Calabi-Yau threefolds contain rational curves (see [Wil], [HW] and [EJS]);(2) C2(X) = 0 for such X but C2(X) ^ 0 for others.Here, for the last statement, we recall the following result due to S. Kobayashi in the smooth case and Shepherd-Barron and Wilson in the general case: THEOREM ([KB, CHAP.IV, COROLLARY (4.15)], [SBW, COROLLARY]).Let X be a Calabi-Yau threefold.Then, X is of Type A if and only if C2(X) = 0. □ One of the main purposes of this paper is to compensate for this defect by revealing explicit geometric structures of Calabi-Yau threefolds of Type A. It will turn out that they are remarkably few so that one can in principle handle them separately in case.Our result is
Character tables are presented for all the 56 non-abelian simple groups of order up to 106, including three tables for the family PSL(2,q) and twenty tables for individual groups (with …
Character tables are presented for all the 56 non-abelian simple groups of order up to 106, including three tables for the family PSL(2,q) and twenty tables for individual groups (with some overlap). Information presented includes the power maps, the orders of the centralizers of elements, and tables of structure constants for each class of involutions.
1. In their fundamental paper of 1949, Higman, Neumann and Neumann proved for the first time that a countable group can always be embedded in some 2-generator group : [1], …
1. In their fundamental paper of 1949, Higman, Neumann and Neumann proved for the first time that a countable group can always be embedded in some 2-generator group : [1], Theorem IV. Two kinds of improvement of this result have recently appeared. In [4], Theorem 2, Dark has shown that the embedding can always be made subnormally . On the other hand, in [2], Theorem 2.1, Levin has shown that the two generators can be given preassigned orders m > 1 and n > 2; and in [3], Miller and Schupp prove that the 2-generator group can also be made to satisfy several additional requirements, such as being complete and Hopfian.
This paper proves explicit formulas for the number of dissections of a convex regular polygon modulo the action of the cyclic and dihedral groups. The formulas are obtained by making …
This paper proves explicit formulas for the number of dissections of a convex regular polygon modulo the action of the cyclic and dihedral groups. The formulas are obtained by making use of the Cauchy–Frobenius Lemma as well as bijections between rotationally symmetric dissections and simpler classes of dissections. A number of special cases of these formulas are studied. Consequently, some known enumerations are recovered and several new ones are provided.
In this paper we effect a systematic study of transitive subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M 24"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> …
In this paper we effect a systematic study of transitive subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M 24"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{M_{24}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, obtaining 5 transitive maximal subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M 24"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{M_{24}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of which one is primitive and four imprimitive. These results, along with the results of the paper, <italic>On subgroups of</italic> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M 24"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{M_{24}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. I, enable us to enumerate all the maximal subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M 24"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{M_{24}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. There are, up to conjugacy, nine of them. The complete list includes one more in addition to those listed by J. A. Todd in his recent work on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M 24"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{M_{24}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The two works were done independently employing completely different methods.
We study Noether's rationality problem for actions of finite groups on projective three-space.
We study Noether's rationality problem for actions of finite groups on projective three-space.
Introduction.The study of a finite 2-transitive group T from a geometric point of view involves intransitive subgroups A displacing all points.For, at least two block designs may be associated with …
Introduction.The study of a finite 2-transitive group T from a geometric point of view involves intransitive subgroups A displacing all points.For, at least two block designs may be associated with such a pair V, A. These are constructed by choosing an orbit B of A, and letting points be the points permuted by Y and blocks be the distinct sets Br, y e I\ In the particular case in which the degree v of T is ¡P:A|, this construction produces two designs, both of which are symmetric designs.That is, there are v points and blocks, A:=|2?| points on each block, k blocks on each point, and, if A=k(k-1 )/(v-I), every two distinct points are on A blocks and every two distinct blocks are on A points.Moreover, Y acts as a 2transitive automorphism group of these symmetric designs, and A is the stabilizer of a block.Conversely, if a symmetric design admits an automorphism group T 2-transitive on points, then T has such a subgroup A of index v, namely, the stabilizer of a block.We are thus led to the problem of determining all symmetric designs admitting 2-transitive automorphism groups.The design of points and hyperplanes of a finite projective space has this property.The unique Hadamard design ^n with »=11, k = 5 and A=2 admits a 2-transitive automorphism group isomorphic to PSL(2, 11) ([31]; see §2).We will place restrictions on symmetric designs and their automorphism groups in order to obtain characterizations of projective spaces and ^n among symmetric designs.However, we will not attempt to classify the groups themselves, as this is an entirely different type of problem [32].Throughout this section, and most of this paper, we will assume that k\(v-1).This condition is satisfied by both projective spaces and J^i-Let Y, A and B be as before, suppose that |T:Aj = v and k\(v-1), and let 2 be the corresponding symmetric design.A has two orbits, B and its complement ^B.Y has a simple normal 2-transitive subgroup Y* such that T* O A is still transitive on both B and ^B; this generalizes a result of Wagner [32] concerning 2-transitive collineation groups of finite projective spaces.If A is not faithful on B then 2 is a finite projective space and Y contains the little projective group ; this is a special case of a result of Ito [17].If A is faithful and 2-transitive on B, then it has a simple normal subgroup which is 2-transitive on B and transitive on ^B.These results, proved in §5, depend heavily on the fact that A is primitive on ^B.
The strong symmetric genus of a group is the smallest genus of a surface on which the group acts faithfully as a group of orientation preserving automorphisms. In this paper …
The strong symmetric genus of a group is the smallest genus of a surface on which the group acts faithfully as a group of orientation preserving automorphisms. In this paper we announce and prove the strong symmetric genus for the hyperoctahedral groups. This parameter is already known for the alternating and symmetric groups as well as the sporadic finite simple groups.
Presentation in terms of generators and relations for the classical finite simple groups of Lie type have been given by Steinberg and Curtis [2,4]. These presentations are useful in proving …
Presentation in terms of generators and relations for the classical finite simple groups of Lie type have been given by Steinberg and Curtis [2,4]. These presentations are useful in proving characterzation theorems for these groups, as in the author's work on the projective symplectic groups [5]. However, in some cases, the application is not quite instantaneous, and an intermediate result is needed to provide a presentation more suitable for the situation in hand. In this paper we prove such a result, for the orthogonal simple groups over finite fields of odd characteristic. In a subsequent article we shall use this to give a characterization of these groups in terms of the structure of the centralizer of an involution.
The holomorph of a group G is Norm B (λ(G)), the normalizer of the left regular representation λ(G) in its group of permutations B = Perm(G). The multiple holomorph of …
The holomorph of a group G is Norm B (λ(G)), the normalizer of the left regular representation λ(G) in its group of permutations B = Perm(G). The multiple holomorph of G is the normalizer of the holomorph in B. The multiple holomorph and its quotient by the holomorph encodes a great deal of information about the holomorph itself and about the group λ(G) and its conjugates within the holomorph. We explore the multiple holomorphs of the dihedral groups D n and quaternionic (dicyclic) groups Q n for n ≥ 3.
Let G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts. We show that a non-cyclic p-group G has …
Let G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts. We show that a non-cyclic p-group G has symmetric genus not congruent to 1(mod p 3) if and only if G is in one of 10 families of groups. The genus formula for each of these 10 families of groups is determined. A consequence of this classification is that almost all positive integers that are the genus of a p-group are congruent to 1(mod p 3). Finally, the integers that occur as the symmetric genus of a p-group with Frattini-class 2 have density zero in the positive integers.
Abstract It is shown that for a finite group G to be isomorphic to a subgroup of SO(3) (or, equivalently, of PSL(2, C) ) it is necessary and sufficient that …
Abstract It is shown that for a finite group G to be isomorphic to a subgroup of SO(3) (or, equivalently, of PSL(2, C) ) it is necessary and sufficient that G satisfies the property that the normalizer of every cyclic subgroup is either cyclic or dihedral.
All finite simple groups of Lie type of rank n over a field of size q , with the possible exception of the Ree groups 2_G_2( q ), have presentations …
All finite simple groups of Lie type of rank n over a field of size q , with the possible exception of the Ree groups 2_G_2( q ), have presentations with at most 49 relations and bit-length O (log n + log q ). Moreover, An and Sn have presentations with 3 generators; 7 relations and bit-length O (log n ), while SL( n , q ) has a presentation with 6 generators, 25 relations and bit-length O (log n + log q ).