Doubling chains, singular elements and hyper-𝒵l-groups

Recognizing simplicity of black-box groups and the frequency of p-singular elements in affine groups

We consider the asymptotic complexity of manipulating matrix groups over finite fields. The question is, given a matrix group G by a list of generators, what can we say in polynomial time about the structure of G? While considerable progress has been made recently in identifying the nonabelian composition factors of a matrix group, the fundamental question of recognizing the simplicity of a non...

Pairwise‎ ‎non-commuting elements in finite metacyclic $2$-groups and some finite $p$-groups

Let $G$ be a finite group‎. ‎A subset $X$ of $G$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $X$ do not commute‎. ‎In this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.

Singular extremals on Lie groups

Chains of Modular Elements and Lattice Connectivity

We show that the order complex of any finite lattice with a chain 0̂ < m1 < · · · < mr < 1̂ of modular elements is at least (r − 2)-connected. In [St], Stanley shows that a finite lattice L with a maximal chain consisting entirely of modular elements is supersolvable and therefore graded and EL-shellable in the sense of [Bj1]. Hence, the order complex ∆(L \ {0̂, 1̂}) has the homotopy type of a wedg...

From singular chains to Alexander Duality