Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. For any fixed prime divisor $p$ of $|G|$, we prove that if every non-nilpotent maximal $A$-invariant subgroup of $G$ of order divisible by $p$ is normal then $G$ has a Sylow tower and $G$ is $q$-nilpotent or $q$-closed for each prime divisor $q$ of $|G|$.
For a finite group $G$ we write $\gamma_\infty(G)$ to denote the nilpotent residual of $G$, that is, the intersection of all terms of the lower central series. The following theorem is proved: Let $G$ be a finite group of odd order admitting an involutory automorphism $\varphi$ such that $G=[G,\varphi]$ and suppose tha...
P. Shumyatsky, Gabriel Santa Catarina de Souza· 0 citations
Let $G$ be a finite nonabelian group of odd order, let $p$ be the smallest prime divisor of $|G|$, and let $V$ be a finite faithful completely reducible $G$-module, possibly of mixed characteristic. Suppose that $M$ is the largest orbit size in the action of $G$ on $V$. It is known that $|G:G'|\le M$, and that equality...
For a fixed prime $p$, a group is called a $p'$-group if the orders of all its elements are coprime to $p$. It is well known that in a finite $p'$-group an automorphism of order $p$ is regular if and only if it is a splitting automorphism of period $p$. In the Kourovka Notebook, V.\,D.~Mazurov asked whether these two n...
We give negative answers to two questions of Farjoun about localizations of groups. For every odd prime $p$, we construct a finite $p$-group $G$, an infinite non-nilpotent group $H$, and a local embedding $f\colon G\hookrightarrow H$. The associated localization satisfies $L_f(G)\cong H$, showing that localizations nee...
Let $q$ be an odd prime power and let $X$ be a maximal curve of genus at least two over $\F_{q^2}$. We prove that the automorphism group of $X$ acts transitively on its rational points if and only if $X$ is the Hermitian curve, or $q=5$ and $X$ is the unique $\F_{25}$-maximal curve of genus $3$, namely the maximal Klei...
Let $G$ be a finite group. We prove that the order of the abelianization $G/G'$ divides the order of $H^{2m}(G,\mathbb{Z})$ for every $m\geq1$. As a consequence, for a field $K$, nonvanishing of $Ext^1_{KG}(K,K)$ implies nonvanishing of $Ext^n_{KG}(K,K)$ for every $n\geq1$. This answers a conjecture by Erdmann, Kl\'asz...
Tiago Cruz, René Marczinzik· 0 citations
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