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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]

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]