Search

A group is called a V-group if it has finite conjugacy classes of subnormal subgroups. It is proved here that if G is a periodic soluble group in which every subnormal subgroup of infinite rank has finitely many conjugates, then G is a V -group, provided that its Hirsch–Plotkin radical has infinite rank. Corresponding results for periodic soluble groups in which every subnormal subgroup of infinite rank has finite index in its normal closure and for those in which every subnormal subgroup of infinite rank is finite over its core, are also obtained. Moreover, it is shown that the assumption on the Hirsch–Plotkin radical can be avoided in the case of periodic groups with nilpotent commutator subgroup. [ABSTRACT FROM AUTHOR]

Abstract: We introduce the notion of an ‘inverse property’ (IP) quandle which we propose as the right notion of ‘Lie algebra’ in the category of sets. For any IP-quandle we construct an associated group . For a class of IP-quandles which we call ‘locally skew’, and when is finite, we show that the noncommutative de Rham cohomology is trivial aside from a single generator θ that has no classical analogue. If we start with a group G then any subset which is ad-stable and inversion-stable naturally has the structure of an IP-quandle. If also generates G then we show that with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group. We prove that this ‘covering map’ is an isomorphism for all finite crystallographic reflection groups W with the set of reflections, and that is locally skew precisely in the simply laced case. This implies that when W is simply laced, proving in particular a conjecture for in Majid (2004) [12]. We also consider as a locally skew IP-quandle ‘Lie algebra’ of and show that , the braid group on 3 strands. The map which therefore arises naturally as a covering map in our theory, coincides with the restriction of the usual universal covering map to the inverse image of . [Copyright &y& Elsevier]

Abstract: It has been proved recently by Moretó (2007) and Craven (2008) that the order of a finite group is bounded in terms of the largest multiplicity of its irreducible character degrees. A conjugacy class version of this result was proved for solvable groups by Jaikin-Zapirain (2005) . In this note, we prove that if G is a finite simple group then the order of G, denoted by , is bounded in terms of the largest multiplicity of its conjugacy class sizes. As a consequence, we prove that if the largest multiplicity of conjugacy class sizes of any quotient of a finite group G is m, then is bounded in terms of m. [Copyright &y& Elsevier]

Abstract: The numbers of connected components of the spectrums of three kinds of rings of characters of a finite group are determined in terms of the numbers of special kinds of conjugacy classes of the group. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study conjugacy classes of the symplectic Renner monoid , and find the following results. Each element in is a join of strictly disjoint cycles and strings. There is an explicit one-to-one correspondence between conjugacy classes and symplectic partitions. A formula for calculating the number of conjugacy classes is obtained. An explicit formula for computing the number of elements in each conjugacy class is given. A relationship between conjugacy classes and Munn conjugacy classes is described. [ABSTRACT FROM AUTHOR]

Abstract: There are well-known monomorphisms between the Artin groups of finite type , and affine type , . The Artin group is isomorphic to the -strand braid group , and the other three Artin groups are isomorphic to some subgroups of . The inclusions between these subgroups yield monomorphisms , and . There are another type of monomorphisms , and which are induced by isomorphisms between Artin groups of type B and centralizers of periodic braids. In this paper, we show that the monomorphisms , and induce injective functions on the set of conjugacy classes, and that none of the monomorphisms , and does so. [Copyright &y& Elsevier]

Letting τ denote the inverse transpose automorphism of GL(n,q), a formula is obtained for the number of g in GL(n,q) so that ggτ is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, ggτ behaves like a hybrid of symplectic and orthogonal groups. It is shown that our formula works well with both cycle index generating functions and asymptotics, and is related to the theory of random partitions. The derivation makes use of models of representation theory of GL(n,q) and of symmetric function theory, including a new identity for Hall–Littlewood polynomials. We obtain information about random elements of finite symplectic groups in even characteristic, and explicit bounds for the number of conjugacy classes and centralizer sizes in the extension of GL(n,q) generated by the inverse transpose automorphism. We give a second approach to these results using the theory of bilinear forms over a field. The results in this paper are key tools in forthcoming work of the authors on derangements in actions of almost simple groups, and we give a few examples in this direction. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]