Related papers: On generations by conjugate elements in almost sim…

Let $\pi$ be a proper subset of the set of all primes. Denote by $r$ the smallest prime which does not belong to $\pi$ and set $m = r$ if $r = 2$ or $3$ and $m = r-1$ if $r \geqslant 5$. We study the following conjecture: a conjugacy class…

Let $\pi$ be a set of primes such that $|\pi|\geqslant 2$ and $\pi$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $\pi$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $r\geqslant 5$. We study the…

A finite group $G$ is \emph{coprimely-invariably generated} if there exists a set of generators $\{g_1, ..., g_u\}$ of $G$ with the property that the orders $|g_1|, ..., |g_u|$ are pairwise coprime and that for all $x_1, ..., x_u \in G$ the…

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…

In this paper we prove that every $2$-generator finite $p$-group $G$ has a non-inner automorphism of order $p$ leaving $G^p\gamma_4(G)$ elementwise fixed ($p\ge 5$). Moreover, we prove a $2$-generator finite $3$-group satisfying…

A longstanding conjecture asserts that every non-abelian finite $p$-group $G$ admits a non-inner automorphism of order $p$. The conjecture is valid for finite $p$-groups of class 2. Here, we prove every finite non-abelian $p$-group $G$ of…

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements…

Let $G$ be a finite simple group of Lie type and let $P$ be a Sylow $2$-subgroup of $G$. In this paper, we prove that for any nontrivial element $x \in G$, there exists $g \in G$ such that $G = \langle P, x^g \rangle$. By combining this…

By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group has a derangement (an element with no fixed points). The existence of derangements with additional properties has attracted much attention,…

As defined by Guralnick and Saxl given a nonabelian simple group $S$ and its nonidentity automorphism $x$, a natural number $\alpha_S(x)$ does not exceed a natural number $m$ if some $m$ conjugates of $x$ in the group $\langle x,S\rangle$…

We study `good elements' in finite $2n$-dimensional classical groups $G$: namely $t$ is a `good element' if $o(t)$ is divisible by a primitive prime divisor of $q^n-1$ for the relevant field order $q$, and $t$ fixes pointwise an $n$-space.…

A finite order element $g$ of a group $G$ is called rational if $g$ is conjugate to $g^i$ for every integer $i$ coprime to the order $g$. We determine all triples $(G,g,\phi)$, where $G$ is a simple algebraic group of type $A_n,B_n$ or…

A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the…

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 $n$…

Every finite simple group can be generated by two elements, and in 2000, Guralnick and Kantor resolved a 1962 question of Steinberg by proving that in a finite simple group every nontrivial element belongs to a generating pair. Groups with…

The proper subgroup $B$ of the group $G$ is called {\it strongly embedded}, if $2\in\pi(B)$ and $2\notin\pi(B \cap B^g)$ for any element $g \in G \setminus B $ and, therefore, $ N_G(X) \leq B$ for any 2-subgroup $ X \leq B $. An element $a$…

A finite group $G$ is called uniformly semi-rational if there exists an integer $r$ such that the generators of every cyclic sugroup $\langle x \rangle$ of $G$ lie in at most two conjugacy classes, namely $x^G$ or $(x^r)^G$. In this paper,…

We prove that every finite simple group of Lie type $G$ can be generated by three regular unipotent elements. In certain cases we show that two regular unipotents are sufficient to generate $G$.

Given a finite abelian group $G$ and elements $x, y \in G$, we prove that there exists $\phi \in \text{Aut}(G)$ such that $\phi(x) = y$ if and only if $G/\langle x \rangle \cong G/\langle y \rangle$. This result leads to our development of…

The minimal degree of a permutation group $G$ is the minimum number of points not fixed by non-identity elements of $G$. Lower bounds on the minimal degree have strong structural consequences on $G$. Babai conjectured that if a primitive…