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

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…

We show that if a finite point set $P\subseteq \mathbb{R}^2$ has the fewest congruence classes of triangles possible, up to a constant $M$, then at least one of the following holds. (1) There is a $\sigma>0$ and a line $l$ which contains…

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.

Let the group $G = AB$ be the product of subgroups $A$ and $B$, and let $p$ be a prime. We prove that $p$ does not divide the conjugacy class size (index) of each $p$-regular element of prime power order $x\in A\cup B$ if and only if $G$ is…

Let $G$ be a finite group, and $\pi$ be a set of primes. The $\pi$-core $\mathbf{O}_\pi(G)$ is the unique maximal normal $\pi$-subgroup of $G$, and $b(G)$ is the largest irreducible character degree of $G$. In 2017, Qian and Yang proved…

The existence of closed orbits of real algebraic groups on real algebraic varieties is established. As an application, it is shown that if G is a real reductive linear group with Iwasawa decomposition G= KAN, then every unipotent subgroup…

Let \pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a…

We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We…

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…

Let $G$ be a finite group, $n$ a positive integer. $\pi(n)$ denotes the set of all prime divisors of $n$ and $\pi(G)=\pi(|G|)$. The prime graph $\Gamma(G)$ of $G$, defined by Grenberg and Kegel, is a graph whose vertex set is $\pi(G)$, two…

The well-known Landau's theorem states that, for any positive integer $k$, there are finitely many isomorphism classes of finite groups with exactly $k$ (conjugacy) classes. We study variations of this theorem for $p$-regular classes as…

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…

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…

Let $G$ be a finite group and let $\pi$ be a set of primes. In this paper, we prove a criterion for the existence of a solvable $\pi$-Hall subgroup of $G$, precisely, the group $G$ has a solvable $\pi$-Hall subgroup if, and only if, $G$ has…

Every finite group whose order is divisible by a prime $p$ has at least $2 \sqrt{p-1}$ conjugacy classes.

Let $G$ be a finite group and assume $p$ is a prime dividing the order of $G$. Suppose for any such $p$, that every two abelian $p$-subgroups of $G$ of equal order are conjugate. The structure of such a group $G$ has been settled in this…

Let $p$ be a prime and let $P$ be a Sylow $p$-subgroup of a finite nonabelian group $G$. Let $bcl(G)$ be the size of the largest conjugacy class of the group $G$. We show that $|P/O_p(G)| < bcl(G)$ if $G$ is not abelian.

This article is about Pi Formulas, infinite series of fractions which sum to multiples of Pi. Each such one can be associated with a unique set $S_k$ of rough numbers, where $k$ is a prime number. Given $S_k$ for any prime $k$, the set…

Suppose that $G$ is a finite solvable group and let $H$ be a Hall $\pi$-subgroup, let $b(H)$ be the largest character degree of $H$, we show that $|G:O_{\pi' \pi}(G)|_{\pi} \leq b(H)^2$.

Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes for any positive integer $k$. We show that, for any positive integers $n$ and $s$, there…