Related papers: On the sharp Baer--Suzuki theorem for $\pi$-radica…

We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group $G$ isospectral to a finite simple group has a unique nonabelian composition factor, that is, the…

A closed subgroup of a semisimple algebraic group is called irreducible if it lies in no proper parabolic subgroup. In this paper we classify all irreducible subgroups of exceptional algebraic groups $G$ which are connected, closed and…

Let $G$ be an almost simple group. We prove that if $x \in G$ has prime order $p \ge 5$, then there exists an involution $y$ such that $<x,y>$ is not solvable. Also, if $x$ is an involution then there exist three conjugates of $x$ that…

It is proved that finite nonabelian simple groups $S$ with $\max \pi(S)=37$ are uniquely determined by their order and degree pattern in the class of all finite groups.

We classify pairs of conjugacy classes in almost simple algebraic groups whose product consists of finitely many classes. This leads to several interesting families of examples which are related to a generalization of the Baer--Suzuki…

If $G$ is a finite group, then the spectrum $\omega(G)$ is the set of all element orders of $G$. The prime spectrum $\pi(G)$ is the set of all primes belonging to $\omega(G)$. A simple graph $\Gamma(G)$ whose vertex set is $\pi(G)$ and in…

Let $p$ be an odd prime, and let $S$ be a $p$-group with a unique elementary abelian subgroup $A$ of index $p$. We classify the simple fusion systems over all such groups $S$ in which $A$ is essential. The resulting list, which depends on…

In representation theory of finite groups, there is a well-known and important conjecture due to M. Brou\'e. He conjectures that, for any prime $p$, if a $p$-block $A$ of a finite group $G$ has an abelian defect group $P$, then $A$ and its…

Given a finite group $G$, let $\pi(G)$ denote the set of all primes that divide the order of $G$. For a prime $r \in \pi(G)$, we define $r$-singular elements as those elements of $G$ whose order is divisible by $r$. Denote by $S_r(G)$ the…

The Pr\"ufer rank $\mathrm{rk}(G)$ of a profinite group $G$ is the supremum, across all open subgroups $H$ of $G$, of the minimal number of generators $\mathrm{d}(H)$. It is known that, for any given prime $p$, a profinite group $G$ admits…

Let G be a finite abelian group with |G|>1. Let a_1,...,a_k be k distinct elements of G and let b_1,...,b_k be (not necessarily distinct) elements of G, where k is a positive integer smaller than the least prime divisor of |G|. We show that…

The partition number $\pi(K)$ of a simplicial complex $K\subset 2^{[n]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus\ldots\uplus A_\nu = [n]$ of $[n]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is…

Many open conjectures in the representation theory of finite groups can be studied by reducing them to related questions about quasi-simple groups. In such studies, $p$-radical subgroups typically play a critical role. To classify the…

Let $G$ be a finite group. Let $k(G)$ denote the number of conjugacy classes of $G$ and let $m(G)$ denote the least positive integer $n$ such that the union of any $n$ distinct non-trivial conjugacy classes of $G$ together with the identity…

We classify finite groups in which the centralisers of certain non-central elements are soluble. This includes a full structural description of groups whose non-central element centralisers are all soluble, and a reduction theorem for the…

We prove that if $\pi$ is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are $\pi$-groups. In particular, when $\pi$ is the empty set, we obtain Henckell's decidability of…

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $\Pi$-property in $G$ if there exists a chief series $\mathit{\Gamma}_G:1=G_0<G_1<\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\leq…

Let $p$ be a prime number. A longstanding conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we prove that if $G$ is an odd order finite non-abelian monolithic $p$-group such…

In this short note we prove that, if $p$ is an odd prime dividing the order of a sporadic simple group, then with the exception of four groups for $p=3$, all sporadic simple groups are generated by an involution and an element of order $p$.

If $G$ is a finite primitive complex reflection group, all reflection subgroups of $G$ and their inclusions are determined up to conjugacy. As a consequence, it is shown that if the rank of $G$ is $n$ and if $G$ can be generated by $n$…