Related papers: Powerful $p$-groups have noninner automorphisms of…

A finite non-abelian group $G$ is called commuting integral if the commuting graph of $G$ is integral. In this paper, we show that a finite group is commuting integral if its central factor is isomorphic to ${\mathbb{Z}}_p \times…

Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groups which are not elementary abelian and have the property that the number of their subgroups is maximal among p-groups of a given order. We complement these results…

Let $\hat G$ be the finite simply connected version of an exceptional Chevalley group, and let $V$ be a nontrivial irreducible module, of minimal dimension, for $\hat G$ over its field of definition. We explore the overgroup structure of…

Let $G$ be a group. An element $g \in G$ is called a test element of $G$ if for every endomorphism $\varphi:G \to G$, $\varphi(g)=g$ implies that $\varphi$ is an automorphism. We prove that for a finitely generated profinite group $G$, $g…

Given a finitely presented group $G$, we wish to explore the conditions under which automorphisms of quotients $G/N$ can be lifted to automorphisms of $G$. We discover that in the case where $N$ is a central subgroup of $G$, the question of…

Given a free group $\Gamma$ of finite rank $n$ and a prime number $p,$ denote by $\Gamma_k^\bullet$ the $k^\text{th}$ layer of the Stallings ($\bullet=S$) or Zassenhaus ($\bullet=Z$) $p$-central series, by $\mathcal{N}_{k}^\bullet$ the…

A loop is said to be automorphic if its inner mappings are automorphisms. For a prime $p$, denote by $\mathcal A_p$ the class of all $2$-generated commutative automorphic loops $Q$ possessing a central subloop $Z\cong \mathbb Z_p$ such that…

Let G be a torsion--free abelian group of finite rank. The automorphism group Aut(G) acts on the set of maximal independent subsets of G. The orbits of this action are the isomorphism classes of indecomposable decompositions of G. G…

Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a faithful, self-similar action on the $p$-ary rooted tree. We prove that if the set $\{g\in G \ | \ g^{p^n}=1\}$ is a nontrivial subgroup for some $n$, then $G$ is a…

For finite p-groups P of class 2 and exponent p the following are invariants of fully refined central decompositions of P: the number of members in the decomposition, the multiset of orders of the members, and the multiset of orders of…

The results of computer computations determining the automorphism groups of the groups of order 32$p$ for $p \geq 3$ are given in several tables. Presentations for the automorphism groups of the groups of order 32, which in many cases…

We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite $p$-groups and finite simple groups with…

We investigate the endomorphism monoids of certain finite $p$-groups of order $p^8$ first studied by Jonah and Konvisser in 1975 as examples for finite $p$-groups with abelian automorphism group, and we show some necessary conditions for a…

Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's…

A rigid automorphism of a linking system is an automorphism which restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that…

Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in…

For a finite group G of Lie type and a prime p, we compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic,…

Let $p$ be an odd prime. Denote a Sylow $p$-subgroup of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ by $S_p(n,GL)$ and $S_p(n,SL)$ respectively. The theory of stable elements tells us that the mod-$p$ cohomology of a finite group is…

In this paper, we construct an infinite families of group $G$ of order $4p^m$ which can be an automorphism group of some regular polytope with type $\{p, 2p\}$, where $m \geq 3$ and $p$ is an odd prime. For $p=3$, our polytopes are the…

Given positive integers $p$ and $m$, where $p$ is assumed to be an odd prime, we determine the automorphism groups of $p$-groups $J$, $H$, and $K$ of orders $p^{7m}$, $p^{6m}$, and $p^{5m}$, and nilpotency classes 5, 4, and 3, respectively,…