In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
In this paper, when the order of $θ$ is even, we prove that there exists no central difference sets in $A_2(m,θ)$ and establish some non-existence results of central partial difference …
In this paper, when the order of $θ$ is even, we prove that there exists no central difference sets in $A_2(m,θ)$ and establish some non-existence results of central partial difference sets in $A_p(m,θ)$ with $p>2$. When the order of $θ$ is odd, we construct central difference sets in $A_2(m,θ)$. Furthermore, we give some reduced linking systems of difference sets in $A_2(m,θ)$ by using the difference sets we constructed. In the case $p>2$, we construct Latin square type central partial difference sets in $A_p(m,θ)$ by a similar method.
In this paper, we completely determine the irreducible characters of the four families of Suzuki [Formula: see text]-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki [Formula: see text]-groups.
Abstract Let be a finite proper partial geometry pg not isomorphic to the van Lint–Schrijver partial geometry pg and let be a group of automorphisms of acting primitively on both …
Abstract Let be a finite proper partial geometry pg not isomorphic to the van Lint–Schrijver partial geometry pg and let be a group of automorphisms of acting primitively on both points and lines of , we show that if then must be almost simple.
Abstract Let be a finite proper partial geometry pg not isomorphic to the van Lint–Schrijver partial geometry pg and let be a group of automorphisms of acting primitively on both …
Abstract Let be a finite proper partial geometry pg not isomorphic to the van Lint–Schrijver partial geometry pg and let be a group of automorphisms of acting primitively on both points and lines of , we show that if then must be almost simple.
In this paper, we completely determine the irreducible characters of the four families of Suzuki [Formula: see text]-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki [Formula: see text]-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
In this paper, when the order of $θ$ is even, we prove that there exists no central difference sets in $A_2(m,θ)$ and establish some non-existence results of central partial difference …
In this paper, when the order of $θ$ is even, we prove that there exists no central difference sets in $A_2(m,θ)$ and establish some non-existence results of central partial difference sets in $A_p(m,θ)$ with $p>2$. When the order of $θ$ is odd, we construct central difference sets in $A_2(m,θ)$. Furthermore, we give some reduced linking systems of difference sets in $A_2(m,θ)$ by using the difference sets we constructed. In the case $p>2$, we construct Latin square type central partial difference sets in $A_p(m,θ)$ by a similar method.
In this note, we construct the irreducible characters of Suzuki p-groups of types A p (m, θ) and C p (m, θ, ϵ).
In this note, we construct the irreducible characters of Suzuki p-groups of types A p (m, θ) and C p (m, θ, ϵ).
Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small …
Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small groups products of representation and characters on the number of solutions gm =1 in a group a theorem of A. Hurwitz on multiplicative sums of squares permutation representations and characters the class number real characters and real representations Coprime action groups pa qb Fronebius groups induced characters Brauer's permutation lemma and Glauberman's character correspondence Clifford theory 1 projective representations Clifford theory 2 extension of characters Degree pattern and group structure monomial groups representation of wreath products characters of p-groups groups with a small number of character degrees linear groups the degree graph groups all of whose character degrees are primes two special degree problems lengths of conjugacy classes R. Brauer's theorem on the character ring applications of Brauer's theorems Artin's induction theorem splitting fields the Schur index integral representations three arithmetical applications small kernels and faithful irreducible characters TI-sets involutions groups whose Sylow-2-subgroups are generalized quaternion groups perfect Fronebius complements. (Part contents).
Abstract A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked …
Abstract A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2‐groups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a linking system of difference sets in non‐2‐groups. We make a connection to Kerdock and bent sets, which provides large linking systems of difference sets in elementary abelian 2‐groups. We give a new construction for linking systems of difference sets in 2‐groups, taking advantage of a previously unrecognized connection with group difference matrices. This construction simplifies and extends prior results, producing larger linking systems than before in certain 2‐groups, new linking systems in other 2‐groups for which no system was previously known, and the first known examples in nonabelian groups.
This paper concerns difference sets in finite groups.The approach is as follows: if D is a difference set in a group G, and χ any character of G, χ(D) -Σχ>x(ί7) …
This paper concerns difference sets in finite groups.The approach is as follows: if D is a difference set in a group G, and χ any character of G, χ(D) -Σχ>x(ί7) is an algebraic integer of absolute value V~n in the field of mth roots of 1, where m is the order of χ.Known facts about such integers and the relations which the χ(D) must satisfy (as χ varies) may yield information about D by the Fourier inversion formula.In particular, if χ(D) is necessarily divisible by a relatively large integer, the number of elements g of D for which χ(g) takes on any given value must be large; this yields some nonexistence theorems.Another theorem, which does not depend on a magnitude argument, states that if n and v are both even and α, the power of 2 in v, is at least half of that in n, then G cannot have a character of order 2 α , and thus G cannot be cyclic.A difference set with v = An gives rise to an Ήadamard matrix; it has been conjectured that no such cyclic sets exist with v > 4.This is proved for n even by the above theorem, and is proved for various odd n by the theorems which depend on magnitude arguments.In the last section, two classes of abelian, but not cyclic, difference sets with v = in are exhibited.A subset D of a finite group G is called a difference set if every element Φe of G can be represented in precisely λ ways as d x d^, d { e D. If χ is any nonprincipal character of G, we must then have | ^jdeD χ(d) \ -Λ/ΊΪ, n -k -λ, where k is the order of D. We shall write χ(D) for ΣidβD lid) (as in [8]).If G is abelian and | χ{D) \ = V~n for some subset D and all nonprincipal characters of G, D is a difference set in G.This work originated in a search for difference sets with G cyclic of order v, and the parameters related by v = in.Because in this case every divisor of n is a divisor of v, Hall's theorem on multipliers, [5], one of the main tools in the study of difference sets, cannot be applied.The method presented here is particularly suitable for computation of difference sets if v and n have common factors.It is roughly as follows: the numbers X(D) are algebraic integers of absolute value V~n in the field of mth roots of 1, where m is the order of χ (as an
Abstract In this paper, we consider regular automorphism groups of graphs in the RT2 family and the Davis‐Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general …
Abstract In this paper, we consider regular automorphism groups of graphs in the RT2 family and the Davis‐Xiang family and amorphic abelian Cayley schemes from these graphs. We derive general results on the existence of non‐abelian regular automorphism groups from abelian regular automorphism groups and apply them to the RT2 family and Davis‐Xiang family and their amorphic abelian Cayley schemes to produce amorphic non‐abelian Cayley schemes.
Abstract Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the …
Abstract Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in where ) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of is a new skew Hadamard difference set in with m odd, where denotes the first kind of Dickson polynomials of order n and . The key observation in the proof is that is a planar function from to for m odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all , the set is a skew Hadamard difference set in , where m is odd and . The proof is more complicated and different than that of Ding‐Yuan skew Hadamard difference sets since is not planar in . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for by comparing the triple intersection numbers.
Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type …
Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type parameters is constructed in nonelementary abelian groups, the groups Z 4 2 k × Z 4 4 l − 4 k for all k when ℓ is odd and for all k < ℓ when ℓ is even. Similarly, partial difference sets with Latin square type parameters are constructed in the same groups for all k when ℓ is even and for all k<ℓ when ℓ is odd. These constructions provide the first example where the non-homomorphic bijection approach outlined by Hagita and Schmidt can produce difference sets in groups that previously had no known constructions. Computer computations indicate that the strongly regular graphs associated to the partial difference sets are not isomorphic to the known graphs, and it is conjectured that the family of strongly regular graphs will be new.
A point in a finite projective plane PG(2, pn), may be denoted by the symbol (xi, x2, x3), where the coordinates xi, x2, x3 are marks of a Galois field …
A point in a finite projective plane PG(2, pn), may be denoted by the symbol (xi, x2, x3), where the coordinates xi, x2, x3 are marks of a Galois field of order pn, GF(pn).The symbol (0, 0, 0) is excluded, and if k is a non-zero mark of the GF(pn), the symbols (xi, Xi, x3) and (kxh kx2, kx3) are to be thought of as the same point.The totality of points whose coordinates satisfy the equation uiXi+u2x2+u3x3 = 0, where ui, w2, u3 are marks of the GF(pn), not all zero, is called a line.The plane then consists of p2n+pn+l = q points and q lines; each line contains pn + \ points, j A finite projective plane, PG(2, pn), defined in this way is Pascalian and Desarguesian ; it exists for every prime p and positive integer », and there is only one such PG(2, pn) for a given p and » (VB, p. 247, VY, p. 151).Let Ao be a point of a given PG(2, pn), and let C be a collineation of the points of the plane.(A collineation is a 1-1 transformation carrying points into points and lines into lines.)Suppose C carries A o into A\, Ax into * Presented to the Society, October 27, 1934, under a different title;
We generalize the definition of Camina groups. We show that our generalized Camina groups are exactly the groups isoclinic to Camina groups, and so many properties of Camina groups are …
We generalize the definition of Camina groups. We show that our generalized Camina groups are exactly the groups isoclinic to Camina groups, and so many properties of Camina groups are shared by these generalized Camina groups. In particular, we show that if G is a nilpotent, generalized Camina group then its nilpotence class is at most 3. We use the information we collect about generalized Camina groups with nilpotence class 3 to characterize the character tables of these groups.
GRAHAM HIGMAN1. Introduction In this paper we shall determine all groups G of order a power of 2 which possess automorphisms that permute their involutions cyclically.The de- termination is complete, …
GRAHAM HIGMAN1. Introduction In this paper we shall determine all groups G of order a power of 2 which possess automorphisms that permute their involutions cyclically.The de- termination is complete, except that we do not exclude the possibility that two or more of the groups that we list may be isomorphic.The investigation is perhaps not without interest simply as an example of the use of linear methods in p-group theory; but the main motivation for it is that some result along these lines is needed by Suzuki in his classification [4] of ZT-groups.It is a pleasure to acknowledge that this paper is, in a direct way, a fruit of the special year in Group Theory organized by the Department of Mathematics at the University of Chicago.A 2-group with only one involution, that is, a eyelie or generalised quaternion group obviously has the property under discussion; and an abelian group has it if and only if it is a direct product of eyelie 2-groups all of the same order.It is convenient to exclude these eases from the beginning, and define a Suzulci 2-group as a non-abelian 2-group with more than one involution, having a eyelie group of automorphisms which permutes its involutions transi- tively.Evidently, the involutions of a Suzuki 2-group G all belong to its center, and so constitute, with the identity, an elementary abelian subgroup fh(G) of order q 2", n > 1.We shall show that fI(G) Z(G) q(G) G', so that G is of exponent 4 and class 2. The automorphism ( which permutes cyclically the q 1 involutions evidently has order divisible by q 1.We shall show that can be taken to have order precisely q 1, and so to be regular.The order of G is either q or qa.In many ways, it would be more satisfactory to impose on G the simpler, weaker condition that the involutions of G are permuted transitively by the full automorphism group of G. Possibly such a relaxation would not bring in any large class of new groups; but the condition seems to be very hard to handle.However, a little of our argument extends to the general ease, and this part has been stated for that ease.The methods used are similar to those involving the associated Lie ring (el.e.g. [2]),but we shall not construct this ring explicitly.The setup, which we shall presuppose, is as follows.If H is a subgroup of the 2-group G, and K a normal subgroup of H with elementary abelian factor group H/K,
Suppose that a group $G$ acts transitively on the points of $\mathcal {P}$, a finite non-Desarguesian projective plane. We prove that if $G$ is insoluble, then $G/O(G)$ is isomorphic to …
Suppose that a group $G$ acts transitively on the points of $\mathcal {P}$, a finite non-Desarguesian projective plane. We prove that if $G$ is insoluble, then $G/O(G)$ is isomorphic to $\mathrm {SL}_2(5)$ or $\mathrm {SL}_2(5).2$.
Abstract Let G be a group of collineations of a finite thick generalised quadrangle Γ. Suppose that G acts primitively on the point set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒫</m:mi> </m:math> ${\mathcal{P}}$ of …
Abstract Let G be a group of collineations of a finite thick generalised quadrangle Γ. Suppose that G acts primitively on the point set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒫</m:mi> </m:math> ${\mathcal{P}}$ of Γ, and transitively on the lines of Γ. We show that the primitive action of G on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒫</m:mi> </m:math> ${\mathcal{P}}$ cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples Γ for which the action of G on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒫</m:mi> </m:math> ${\mathcal{P}}$ is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where there is a unique minimal normal subgroup M and M is non-Abelian.
We give the construction of a partial geometry with parameters s = 4, t = 17, σ = 2. We also obtain two new strongly regular graphs.
We give the construction of a partial geometry with parameters s = 4, t = 17, σ = 2. We also obtain two new strongly regular graphs.
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
In this paper, we completely determine the irreducible characters of the four families of Suzuki $p$-groups.
We prove a structure theorem for a class of finite transitive permutation groups that arises in the study of finite bipartite vertex-transitive graphs. The class consists of all finite transitive …
We prove a structure theorem for a class of finite transitive permutation groups that arises in the study of finite bipartite vertex-transitive graphs. The class consists of all finite transitive permutation groups such that each non-trivial normal subgroup has at most two orbits, and at least one such subgroup is intransitive. The theorem is analogous to the O'Nan-Scott Theorem for finite primitive permutation groups, and this in turn is a refinement of the Baer Structure Theorem for finite primitive groups. An application is given for arc-transitive graphs.
This is the fifth volume of a comprehensive and elementary treatment of finite p -group theory. Topics covered in this volume include theory of linear algebras and Lie algebras. The …
This is the fifth volume of a comprehensive and elementary treatment of finite p -group theory. Topics covered in this volume include theory of linear algebras and Lie algebras. The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.
Abstract A pseudo‐hyperoval of a projective space , q even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo‐hyperoval …
Abstract A pseudo‐hyperoval of a projective space , q even, is a set of subspaces of dimension such that any three span the whole space. We prove that a pseudo‐hyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles that admit a point‐primitive, line‐transitive automorphism group with a point‐regular abelian normal subgroup. Specifically, we show that is flag‐transitive and isomorphic to , where is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).
This is the fourth volume of a comprehensive and elementary treatment of finite p-group theory. As in the previous volumes, minimal nonabelian p- groups play an important role. Topics covered …
This is the fourth volume of a comprehensive and elementary treatment of finite p-group theory. As in the previous volumes, minimal nonabelian p- groups play an important role. Topics covered in this volume include: subgroup structure of metacyclic p- groups Ishikawa's theorem on p- groups with two sizes of conjugate classes p- central p- groups theorem of Kegel on nilpotence of H p -groups partitions of p- groups characterizations of Dedekindian groups norm of p- groups p- groups with 2-uniserial subgroups of small order The book also contains hundreds of original exercises and solutions and a comprehensive list of more than 500 open problems. This work is suitable for researchers and graduate students with a modest background in algebra.
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The group PGL$(2,q)$, $q=p^n$, $p$ an odd prime, is $3$-transitive on the projective line and therefore it can be used to construct $3$-designs. In this paper, we determine the sizes …
The group PGL$(2,q)$, $q=p^n$, $p$ an odd prime, is $3$-transitive on the projective line and therefore it can be used to construct $3$-designs. In this paper, we determine the sizes of orbits from the action of PGL$(2,q)$ on the $k$-subsets of the projective line when $k$ is not congruent to $0$ and 1 modulo $p$. Consequently, we find all values of $\lambda$ for which there exist $3$-$(q+1,k,\lambda)$ designs admitting PGL$(2,q)$ as automorphism group. In the case $p\equiv 3$ mod 4, the results and some previously known facts are used to classify 3-designs from PSL$(2,p)$ up to isomorphism.
Linked systems of symmetric designs are equivalent to 3-class Q-antipodal association schemes. Only one infinite family of examples is known, and this family has interesting origins and is connected to …
Linked systems of symmetric designs are equivalent to 3-class Q-antipodal association schemes. Only one infinite family of examples is known, and this family has interesting origins and is connected to important applications. In this paper, we define linking systems, collections of difference sets that correspond to systems of linked designs, and we construct linking systems in a variety of nonelementary abelian groups using Galois rings, partial difference sets, and a product construction. We include some partial results in the final section.