Narrow Results

Let $G$ and $A$ be finite groups of relative coprime orders and $A$ act on $G$ via automorphisms. An $A$-invariant subgroup $S$ of $G$ is called an $A$-$p$-Sylowizer of an $A$-invariant $p$-subgroup $R$ if $S$ is maximal in $G$ with respect to having $R$ as its Sylow $p$-subgroup. In this paper, we obtain some criteria for $p$-nilpotency of a finite group by analyzing the relation to specific subsets and characteristics of the maximal $A$-invariant subgroups of an $A$-invariant Sylow $p$-subgroup.

Let $A$ be a finite group which acts coprimely on finite group $G$ via automorphisms. We investigate the influence of the structural conditions of the Sylow $2$-subgroups of the fixed point subgroup of the action, $C_G(A)$, working on the non-simplicity or solvability of $G$.

Let G and A be finite groups of relative coprime orders and A act on G via automorphisms. In this paper, we prove that when every maximal A-invariant subgroup of G that contains the normalizer of some Sylow subgroup has prime index, then G is supersolvable; if every non-nilpotent maximal A-invariant subgroup of G has prime index or is normal in G, then G is a Sylow tower group.

Let $G$ and $A$ be finite groups of coprime orders and assume that $A$ acts on $G$ by automorphisms. We investigate groups $G$ in which each non-central $A$-invariant subgroup is self-centralizing. Moreover, we describe the structure of groups $G$ in which every $A$-invariant self-centralizing subgroup is either subnormal or a TI-subgroup of $G$.

Let a finite group $A$ act on a finite group $G$ via automorphism and let $s$ be a positive integer. In this paper, we introduce the probabilistic zeta function of ordered $s$-tuples from $G$ which $A$-generate $G$, denoted by ${\mathrm{P}}^A(G,s)$. When $(|A|,|G|)=1$, it is proved that $G$ is solvable if and only if ${\mathrm{P}}^A(G,s)$ has an Euler product expansion with all factors of the form $1-c/q^s$, where each $q$is a prime power. This is a generalization of Detomi and Lucchini's result in 2003.

If a q-group A acts coprimely on G, we would like to understand how the group theoretical properties of the fixed point subgroup C = C_G(A) are reflected in the set of irreducible characters of G, and vice-versa.