We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and pq, for prime numbers p, q with q|p − 1. We also show that certain twisting deformation of the symmetric group is simple as a Hopf algebra. On the other hand, we p...

We find the nonabelian finite simple groups with order prime divisors not exceeding 1000. More generally, we determine the sets of nonabelian finite simple groups whose maximal order prime divisor is a fixed prime less than 1000. Our results are based on calculations in the computer algebra system GAP.

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...

The recently measured unexpected neutrino mixing patterns have caused a resurgence of interest in the study of finite flavor groups with twoand three-dimensional irreducible representations. This paper details the mathematics of the two finite simple groups with such representations, the Icosahedral group A5, a subgroup of SO(3), and PSL2(7), a subgroup of SU(3). ∗E-mail: [email protected] †E-m...

Let G be a finite group and let X be a conjugacy class of G. The rank of X in G, denoted by rank(G:X) is defined to be the minimal number of elements of X generating G. In this paper we review the basic results on generation of finite simple groups and we survey the recent developments on computing the ranks of finite simple groups.

How can we describe all finite groups? Before we address this question, let’s write down a list of all the finite groups of small orders ≤ 15, up to isomorphism. We have seen almost all of these already. If G is abelian, it is easy to write down all possible G of a given order, using the Fundamental Theorem of Finite Abelian Groups: G must be isomorphic to a direct product of cyclic groups, and...

We prove that generous non-nilpotent Borel subgroups of connected minimal simple groups of finite Morley rank are self-normalizing. We use this to introduce a uniform approach to the analysis of connected minimal simple groups of finite Morley rank through a case division incorporating four mutually exclusive classes of groups. We use these to analyze Carter subgroups and Weyl groups in connect...