Nilpotent residual and fitting subgroup of fixed points in finite groups

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup

Fixed Points of Finite Groups Acting on Generalised Thompson Groups

We study centralisers of finite order automorphisms of the generalised Thompson groups Fn,∞ and conjugacy classes of finite subgroups in finite extensions of Fn,∞. In particular we show that centralisers of finite automorphisms in Fn,∞ are either of type FP∞ or not finitely generated. As an application we deduce the following result about the Bredon type of such finite extensions: any finite ex...

On the Length of Finite Groups and of Fixed Points

The generalized Fitting height of a finite group G is the least number h = h∗(G) such that F ∗ h (G) = G, where the F ∗ i (G) is the generalized Fitting series: F ∗ 1 (G) = F ∗(G) and F ∗ i+1(G) is the inverse image of F ∗(G/F ∗ i (G)). It is proved that if G admits a soluble group of automorphisms A of coprime order, then h∗(G) is bounded in terms of h∗(CG(A)), where CG(A) is the fixed-point s...

On $m^{th}$-autocommutator subgroup of finite abelian groups

Let $G$ be a group and $Aut(G)$ be the group of automorphisms of‎ ‎$G$‎. ‎For any natural‎ number $m$‎, ‎the $m^{th}$-autocommutator subgroup of $G$ is defined‎ ‎as‎: ‎$$K_{m} (G)=langle[g,alpha_{1},ldots,alpha_{m}] |gin G‎,‎alpha_{1},ldots,alpha_{m}in Aut(G)rangle.$$‎ ‎In this paper‎, ‎we obtain the $m^{th}$-autocommutator subgroup of‎ ‎all finite abelian groups‎.

Relative non-Normal Graphs of a Subgroup of Finite Groups

Let G be a finite group and H,K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H , denoted by NH,K, which is a bipartite graph with vertex sets HHK and KNK(H) and two vertices x ∈ H HK and y ∈ K NK(H) are adjacent if xy / ∈ H, where HK =Tk∈K Hk and NK(H) = {k ∈ K : Hk = H}. We determined some numerical invariants and state that when this graph is planar or...