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In this paper we study non-solvable and non-Frobenius Camina pairs (G,N). It is known [D. Chillag, A. Mann, C. Scoppola, Generalized Frobenius groups II, Israel J. Math. 62 (1988) 269–282] that in this case N is a p-group. Our first result (Theorem 1.3) shows that the solvable residual of G/Op(G) is isomorphic either to SL(2,pe),p is a prime or to SL(2,5), SL(2,13) with p=3, or to SL(2,5) with p⩾7.Our second result provides an example of a non-solvable and non-Frobenius Camina pair (G,N) with |Op(G)|=55 and G/Op(G)≅SL(2,5). Note that G has a character which is zero everywhere except on two conjugacy classes. Groups of this type were studies by S.M. Gagola [S.M. Gagola, Characters vanishing on all but two conjugacy classes, Pacific J. Math. 109 (1983) 363–385]. To our knowledge this group is the first example of a Gagola group which is non-solvable and non-Frobenius.

In [A. Turull, J. Algebra 235 (2001), 275–314], we calculated the Schur index of each of the irreducible characters of the finite special linear groups. In the present paper, we calculate the Schur index of all the irreducible characters of some overgroups of the special linear groups. The overgroups in question are the special linear groups extended by diagonal automorphisms, and the subgroups of the general linear group that contain the special linear group. To each conjugacy class of irreducible characters of the special linear group in each overgroup is associated a Clifford class. The Clifford class controls all the irreducible characters of the overgroup and intermediate subgroups that are related to the given irreducible by Clifford theory. Knowing only the Clifford class, we can parametrize all the irreducible characters of the intermediate subgroups, and compute, for each parametrized irreducible character, its field of character values, as well as its Schur index over each field. We explicitly compute the Clifford class in each case, and deduce from it the information on the Schur index of all the irreducible characters of the overgroups.

In this paper we are going to give an answer to the following two questions. Let K be a field and A4 the space of n x n square matrices with coefficients in K. The first question we want to investigate is what are the equations of the subvariety A’, of ikf, consisting of rank h matrices A such that A2 = A. By equations we mean of course a set of explicit generators of the prime ideal of x,. The second question is what are the equations of the varieties Yh, h

In general, groups obtained by adjoining roots to conjugacy separable groups need not be conjugacy separable. We derive some conditions so that, by adjoining roots to certain classes of conjugacy separable groups, the resulting groups are again conjugacy separable. We apply this result to show that all residually finite 1-relator groups of the form 〈a, b; a−αbβaαbγ〉 are conjugacy separable. We also show that groups of the form 〈a, b; (a−αblaαbm)s〉, where s > 1, are conjugacy separable.

It is well known that Seifert 3-manifold groups are residually finite. Niblo [J. Pure Appl. Algebra 78 (1992) 77–84] showed that they are double coset separable. Applying this result we show that, except for some special cases, most of the Seifert 3-manifold groups are conjugacy separable.

We show that, up to isomorphism, there are ten connected locally (M11,2)-arc-transitive graphs. We compute, up to conjugacy, the connected locally (G,s)-arc-transitive graphs for G=M11, M12, M22, M23, M24, J1, J2, J3, HS, McL, He, Ru, Suz, Co3, and we obtain a partial classification for O′N. The classification provides locally (G,7)- and (G,9)-arc-transitive graphs for He and Ru, and locally (G,7)-arc-transitive graphs for J3 and O′N. We also obtain new cubic 5-arc-transitive graphs for He and Ru.

In earlier work, the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow $p$-subgroup $U(q)$ of a finite Chevalley group $G(q)$, valid when $q$ is a power of a good prime for $G(q)$. In this paper we develop this algorithm and discuss an implementation in the computer algebra language {\sf GAP}. Using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in $U(q)$, when $G(q)$ is of rank at most 6. In these cases, we observe that the number of conjugacy classes of $U(q)$ is given by a polynomial in $q$ with integer coefficients., 14 pages: Significant revisions and explanations expanded

The sporadic simple groups F 22 and F 23 of orders 2 17 · 3 9 · 5 2 · 7 · 11 · 13 and 2 18 · 3 13 · 5 2 · 7 · 11 · 13 · 17 · 23, respectively, are defined by Fischer. Each is generated by a conjugacy class of involutions called 3-transpositions, any noncommuting pair of which has product of order 3. In another article we describe the group F 22 in terms of a subgroup S 10 by assigning names to each of the 3510 transpositions which reflect how the S 10 acts on them, and we compute a full transform table listing a b for any two transpositions a and b . In this paper the process is extended to describe the group F 23 in terms of a subgroup S 12 . We determine how the S 12 acts on the generating transpositions of F 23 and assign names to them which reflect this action. We then determine which pairs of transpositions commute, and when transpositions a and b do not commute we compute the conjugate a b , thus completing the transform table. The work involved in this paper is similar to that of our other paper, though it is considerably more complex. We rely heavily on its techniques and results, to which constant reference is made.

For a finite group G and a fixed prime p, one can attach to each irreducible Brauer character φ of G a p-subgroup Q, called the vertex of φ, that is unique up to conjugacy. In this paper we examine the behavior of the vertices of φ with respect to normal subgroups when G is assumed to be p-solvable. For arbitrary finite p-solvable groups, we develop a correspondence between the set of Brauer characters of G with vertex Q and the set of Brauer characters of a certain subgroup of G with vertex Q. Moreover, in the case that G has odd order, we extend a result of Navarro regarding the behavior with respect to normal subgroups of a correspondence of Brauer characters of p ′ -degree to a result regarding the behavior with respect to normal subgroups of a correspondence of Brauer characters of arbitrary degree.

In this thesis we classify the conjugacy classes of involutions in Aut g, where g is an affine Kac-Moody Lie Algebra. We distinguish between two kinds of involutions, those which preserve the conjugacy class of a Borel subalgebra and those which don't. We give a complete and non-redundant list of representatives of involutions of the first kind and we compute their fixed points sets. We prove that any involution of the first kind has a conjugate which leaves invariant the components of the Gauss decomposition g = n− ⊕h⊕n+. We also give a complete list of representatives of the conjugacy classes of involutions of the second kind.

A group A is said to be conjugacy separable (c.s.) if for each pair of elements x, y ϵ A such that x, y are not conjugate in A there exists a finite homomorphic image A of A such that the images of x, y in A are not conjugate in A . We show that certain special polyhedral groups and 1-relator groups with torsion of the form 〈b, t; (t−1bltbm)s〉, s >1, are c.s. The latter partly answers a question raised by Anshel in Math. Rev. 81f Review 20038.

We study conjugacy closed loops (CC-loops) and power-associative CC-loops (PACC-loops). If $Q$ is a PACC-loop with nucleus $N$, then $Q/N$ is an abelian group of exponent 12; if in addition $Q$ is finite, then $|Q|$ is divisible by 16 or by 27. There are eight nonassociative PACC-loops of order 16, three of which are not extra loops. There are eight nonassociative PACC-loops of order 27, four of which have the automorphic inverse property. We also study some special elements in loops, such as Moufang elements, weak inverse property (WIP) elements, and extra elements. In a CC-loop, the set of WIP and the set of extra elements are normal subloops. For each $c$ in a PACC-loop, $c^3$ is WIP, $c^6$ is extra, and $c^{12} \in N$., 31 pages, 12pt amsart; v.3: further revisions after second round of refereeing

Key to a computational study of the finite classical groups in odd characteristic are efficient methods for constructing involutions and their centralisers. Constructing an involution in a given conjugacy class is usually achieved by finding an element of even order that powers up to an involution in the class. Lower bounds on the proportions of such elements are therefore required to control the failure probability of these algorithms. Previous results of Christopher Parker and Robert Wilson give an O ( n − 3 ) lower bound that holds for all involution classes in n-dimensional simple classical groups in odd characteristic. We improve this lower bound to O ( n − 2 log n ) , and in certain cases to O ( n − 1 ) , and also treat a larger family of (not necessarily simple) classical groups.

A subgroup H of a periodic group G is said to be Sylow permutable in G if HS=SH for every Sylow subgroup S of G. Like normality, Sylow permutability is not a transitive relation. In this paper we characterize periodic locally soluble groups with Chernikov conjugacy classes (periodic locally soluble CC-groups) in which Sylow permutability is a transitive relation (PST-groups) describing their structure in a very detailed way then extending the structure of finite soluble PST-groups. Moreover we give an effective construction of these groups.

For g a split semisimple Lie algebra over a field of characteristic 0, we link locally trivial principal homogeneous spaces over Spec(R) to the question of conjugacy of maximal abelian diagonalizable subalgebras of g(R)., 18 pages, to appear in Journal of Algebra

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a ``matrix problem''. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GL_n(k) for n < 12 and k a perfect field. If our field is finite of size q, we also show that the number of conjugacy classes, and so the number of characters, of these groups is a polynomial in $q$ with integral coefficients., 23 pages, 6 figures. See also http://zaphod.uchicago.edu/~murray/research/index.html . Submitted to Journal of Algebra

A known result for the finite general linear group GL ( n , F q ) and for the finite unitary group U ( n , F q 2 ) posits that the sum of the irreducible character degrees is equal to the number of symmetric matrices in the group. Fulman and Guralnick extended this result by considering sums of irreducible characters evaluated at an arbitrary conjugacy class of GL ( n , F q ) . We develop an explicit formula for the value of the permutation character of U ( 2 n , F q 2 ) over Sp ( 2 n , F q ) evaluated at an arbitrary conjugacy class and use results concerning Gelfand–Graev characters to obtain an analogous formula for U ( n , F q 2 ) in the case where q is an odd prime. These results are also given as probabilistic statements.

We discuss mixing and covering theorems in the symmetric groups. We present an example of a covering without mixing, and study the conjugacy class [2 n /2 ] of symmetric group S n , which demonstrates mixing without covering. We derive some new character identities from the computation of [2 n /2 ] 2 , and also compute [2 n /2 ] 3 , filling a hole between theorems of J. L. Brenner ( J. Austral. Math. Soc. Ser. A 25, 1978, 210–214) and Y. Dvir ( in “Products of Conjugacy Classes in Groups,” Lecture Notes in Math., Vol. 1112, Springer-Verlag, New York/Berlin, 1985). This computation also motivates a certain classification of 3-colored 3-regular graphs.

Let G be a classical group over the field with q = p d elements. We prove that if p is an odd prime then the irreducible characters of the Sylow p -subgroup of G have degrees that are powers of q . Moreover, we prove that if p is odd and G is not the unitary group then the Borel subgroup of G is an M -group and its character degrees as well as the sizes of the conjugacy classes can be written in the form q r ( q −1) s 2 k .

The study of asymptotic properties of the conjugacy class of a random element of the finite affine group leads one to define a probability measure on the set of all partitions of all positive integers. Four different probabilistic understandings of this measure are given—three using symmetric function theory and one using Markov chains. This leads to non-trivial enumerative results. Cycle index generating functions are derived and are used to compute the large dimension limiting probabilities that an element of the affine group is separable, cyclic, or semisimple and to study the convergence to these limits. The semisimple limit involves both Rogers–Ramanujan identities. This yields the first examples of such computations for a maximal parabolic subgroup of a finite classical group.

We study existentially closed CSA-groups. We prove that existentially closed CSA-groups without involutions are simple and divisible, and that their maximal abelian subgroups are conjugate. We also prove that every countable CSA-group without involutions embeds into a finitely generated one having, up to conjugacy, the same maximal abelian subgroups, except maybe the infinite cyclic ones. We deduce from this that there exist 2 ℵ 0 countable existentially closed CSA-groups without involutions and that their first-order theories have 2 ℵ 0 types over ∅.

The Garside group, as a generalization of braid groups and Artin groups of finite types, is defined as the group of fractions of a Garside monoid. We show that the semidirect product of Garside monoids is a Garside monoid. We use the semidirect product $\mathbb Z\ltimes G^n$ of the infinite cyclic group $\mathbb Z$ and the cartesian product $G^n$ of a Garside group $G$ to study the properties of roots and powers of elements in $G$. The main result is an estimate of the growth of the minimal word-length of powers of elements in Garside groups, when the generating set is the set of simple elements. A direct application is that the set of translation numbers in Garside groups is discrete. It gives an affirmative answer to the question of Gersten and Short for the case of Garside groups. The original question is for biautomatic groups. And we show that the root extraction problem in a Garside group $G$ can be reduced to a conjugacy problem in $\mathbb Z\ltimes G^n$. Using the algorithm for the conjugacy problem in Garside groups, the root extraction problem is decidable for any Garside group., Title has been changed from "Growth of minimal word-length in Garside groups" To appear in Journal of Algebra

In this paper we show using a purely combinatorial argument that a finitely generated infinite group such that fE(n)⩽ans, where a is a constant, admits for every ϵ a sequence {gi,ϵ} of non-unit elements whose centralizer contains more than i1/2−ϵ elements of length less than i. Of course, the interest of this result is in the fact that it excludes the possibility that the group is a pure torsion group, since otherwise the existence of the sequence {gi,ϵ} is obvious. As an application of this result, we show that, in the case where r

Recently, Thompson [9] constructed a new simple group E of order 215 ∘ 3105372 ∘ 13 ∘ 19 ∘ 31 = 90, 745, 943, 887, 872, 000. In particular, E contains only one conjugacy class of involutions, and if e is an involution in E then CE(e) is a (nonsplit) extension of an extra-special 2-group of order 29 by A 9, the alternating group of degree 9. The aim of this paper is to prove the following result.

The series “Weakly closed p-elements and odd automizers” [l IA-1 1 C] dealt with groups G in which a Sylow p-normalizer is “essentially” an odd-order Frobenius group with a non-trivial Schur multiplier. This sequel explores the analogous situation in which the Frobenius group has an even-order complement; in this case the desired conclusion is not p-length 1, but rather a restriction on the number of conjugacy classes of involutions in G. The main result is

LetG be a finite group and θ a complex character ofG. Define Irr(θ) to be the set of all irreducible constituents of θ andIrr(G) to be the set of all irreducible characters ofG. Thecharacter-covering number of a finite groupG, ccn(G), is defined as the smallest positive integer m such thatIrr(χm) =Irr(G) for allχ∈Irr(G)—{1G}. If no such positive integer exists we say that the character-covering-number ofG is infinite. In this article we show that a finite nontrivial groupG has a finite character-covering-number if and only ifG is simple and non-abelian and ifG is a nonabelian simple group thenccn(G) ⩽ k2 − 3k + 4, wherek is the number of conjugacy classes ofG. Then we show (using the classification of the finite simple groups) that the only finite group with a character-covering-number equal to two is the smallest Janko's group,J1. These results are analogous to results obtained previously concerning the covering of groups by powers of conjugacy classes. Other related results are shown.

For Bernd Fischer on the occasion of his 70th birthdayIntroduction: When Gis a nite group, we may endow G Gwith the structureof a probability space by assigning the uniform distribution. As was pointed outby W.H. Gustafson [10], the probability that a randomly chosen pair of elementsof Gcommute is then

In earlier work it was shown that each nonabelian finite simple group G has a conjugacy class C such that, whenever 1 ≠ x ∈ G , the probability is greater than 1/10 that G = 〈 x , y 〉 for a random y ∈ C . Much stronger asymptotic results were also proved. Here we show that, allowing equality, the bound 1/10 can be replaced by 13/42; and, excluding an explicitly listed set of simple groups, the bound 2/3 holds. We use these results to show that any nonabelian finite simple group G has a conjugacy class C such that, if x 1 , x 2 are nontrivial elements of G, then there exists y ∈ C such that G = 〈 x 1 , y 〉 = 〈 x 2 , y 〉 . Similarly, aside from one infinite family and a small, explicit finite set of simple groups, G has a conjugacy class C such that, if x 1 , x 2 , x 3 are nontrivial elements of G, then there exists y ∈ C such that G = 〈 x 1 , y 〉 = 〈 x 2 , y 〉 = 〈 x 3 , y 〉 . We also prove analogous but weaker results for almost simple groups.

Let G be the group PSL n ( F ), where n ≥3, F is a field, and | F |≥4. Assume, further, that if n =3, then F is either finite or algebraically closed. Given an integer k and a subset A ⊆ G , denote A k ={ a 1 a 2 ··· a k | a 1 , a 2 ,…, a k ∈ A }. Denote by cn( G ) the minimal value of k such that C k = G for every nontrivial conjugacy class C of G . It is shown that cn( G )= n . Related results on factorizations of matrices in GL n ( F ) and SL n ( F ) are also discussed.

Let G be a nonsolvable subgroup of Diffω(C, 0). We derive an infinite sequence of additive subgroups Gk ⊂ C, k ∈ N, depending only on the conjugacy class of G within Diffω(C, 0). As an application, we prove that G is conjugate within Diffω(C, 0) to a subgroup of Diffω(R, 0) if and only if all Gk are real.

Scott (1978) [12] showed Seifert 3-manifold groups are subgroup separable. Niblo (1992) [9] improved this result by showing that these groups are double coset separable. In Allenby, Kim and Tang (2005) [2] it was shown that all but two types of groups in the orientable case are conjugacy separable. Martino (2007) [7] using topological results showed that Seifert groups are conjugacy separable. Here we use algebraic method to show that Seifert groups over non-orientable surfaces are conjugacy separable.

Let G be a finite group and k be a field whose characteristic p divides the order of G. In [3] we introduce the notion of Auslander-Reiten system of G on a symmetric interior G-algebra A (that is a symmetric k-algebra together with a homomorphism 4: G + A x ), as a generalization of the notion of almost split sequences of kG-modules. The language of interior G-algebras proves to be useful to study their restriction to certain subgroups such as the defect group (vertex) of their extremity. Furthermore we show in [3] that each non projective primitive idempotent i of AG (see Section 1) is the extremity of a unique Auslander-Reiten system, up to embedding of A into other symmetric interior G-algebras and to conjugacy (cf. [3, VI], recall embeddings are those one-to-one homomorphisms f: A + B which satisfy Im f = f (1) Af (1)). Thus in particular the middle term of “the” Auslander-Reiten system terminating in i is well defined (up to embedding and conjugacy), and we may look at its decomposition into primitive idempotents. Now if we set A [G] = A @ kG, we have a symmetric algebra again, which is similar to kG in many ways, and which enables us to view our systems of G over A ([3, I]) as short exact sequences of modules: to any idempotent i of AG, we associate the A-projective A[G]-module iA (with (b@ g) . iu = d(g) iub). This way any system Y = (i, i”, i’, d, d’) of G over A determines an exact sequence of A[G]-modules

We give an upper bound for the number of conjugacy classes of closed subgroups of the full wreath product FWrWSym(Ω) which project onto Sym(Ω). Here, Ω is infinite, W is the set of n-tuples of distinct elements from Ω (for some finite n), F is a finite nilpotent group, and the topology on the wreath product is that of pointwise convergence in its imprimitive permutation action. The result addresses a problem which arises in a natural model-theoretic context about classifying certain types of finite covers.

We prove a generalization of the Brauer–Nesbitt theorem. We study the Robinson map and relations between p-regular conjugacy classes and block idempotents with common defect group. We characterize defect classes and those class sums whose images under the Brauer map are not nilpotent.

Two character tables of finite groups are isomorphic if there exist a bijection for the irreducible characters and a bijection for the conjugacy classes that preserve all the character values. We give necessary and sufficient conditions for two finite groups to have isomorphic character tables. In the case of finite p-groups with derived subgroup of order p, we show that the character tables can be classified by equivalence classes of certain homomorphisms of abelian p-groups.

We continue our study of the characters of the Weyl groups of the simple Lie algebras, begun in [2], in order to find a unified theory using their common structure as reflection groups. Carter [1] dealt with the same problem for the conjugacy classes, and we will adopt his use of Weyl subgroups. In this paper we look at the Weyl group of type C or hyper-octahedral group—the wreath product C 2 ≀ S l . We first construct the irreducible characters from those of the symmetric group and, like the conjugacy classes, they will be parameterized by pairs of partitions. We shall then be able to give an algorithm generalizing the partial ordering on partitions given in [2], which will allow us to determine the irreducible constituents of the principal character of a Weyl subgroup induced up to the whole group.

Let G c Aut,( V) be a subgroup of automorphisms of a finite-dimensional vector space V over a field k. A vector u E I’ is said to be a G-orbit basis generator for V if the set G(u) is a linear basis for I’. In this paper we continue the study begun in [Rl, R2] of the following question: If U, U are G-orbit basis generators for V then when are the stabilizers G, and G, conjugate? The conjugacy question is a more general formulation of a question concerning finite-dimensional separable field extensions E, F over an infinite field k of characteristic p > 0. Let k be a field of characteristic p > 0 and suppose E and F are n-dimensional separable field extensions of k. Recall that the associative structure of any commutative algebra A over k affords A a restricted Lie (Frobenius) structure Y(A) by [ab] =0 and aCpl =up for a, h E A. A natural question to ask about the pair E and F is the following: If Z(E) and Z(F) are isomorphic Lie algebras, then when are E and F isomorpic field extensions? Suppose P(E) N P(F). It does not necessarily follow that E = F

Let G be a finite solvable group. We assume that the set of conjugacy class sizes of G is {1, m, n, m k} with m and n coprime positive integers greater than 1 and k a divisor of n. Then we obtain several properties on the structure of G. © 2007 Elsevier Inc. All rights reserved.

In this paper we prove the following long-standing conjecture in the theory of finite groups: Finite solvable groups with no two distinct conjugacy classes of the same length are isomorphic to the symmetric group of degree 3.