Cutoff for product replacement on finite groups

On the product decomposition conjecture for finite simple groups

We prove that if G is a finite simple group of Lie type and S a subset of G of size at least two then G is a product of at most c log |G|/ log |S| conjugates of S, where c depends only on the Lie rank of G. This confirms a conjecture of Liebeck, Nikolov and Shalev in the case of families of simple groups of bounded rank. We also obtain various related results about products of conjugates of a s...

ON THE CHARACTERISTIC DEGREE OF FINITE GROUPS

In this article we introduce and study the concept of characteristic degree of a subgroup in a finite group. We define the characteristic degree of a subgroup H in a finite group G as the ratio of the number of all pairs (h,α) ∈ H×Aut(G) such that h^α∈H, by the order of H × Aut(G), where Aut(G) is the automorphisms group of G. This quantity measures the probability that H can be characteristic ...

A finite basis theorem for product varieties of groups

It is shown that, if IJ is a subvariety of the join of a nilpotent variety and a metabelian variety and if V̂ is a variety with a finite basis for its laws, then UV also has a finite basis for its laws. The special cases IJ nilpotent and IJ metabelian have been established by Higman (1959) and Ivanjuta (1969) respectively. The proof here, which is independent of Ivanjuta's, depends on a rather g...

Properties of representative product systems on the complete product of non-commutative finite groups

Growth in Product Replacement Graphs of Grigorchuk Groups

The product replacement graph Γk(G) is the graph on the generating k-tuples of a group G, with edges corresponding to Nielsen moves. We prove the exponential growth of product replacement graphs Γk(Gω) of Grigorchuk groups, for k ≥ 5.