### $1{\textstyle \frac{1}{2}}$-generation of finite simple groups.

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2010 Mathematics Subject Classification: 20F05, 20D06.We prove that the group PSL6(q) is (2,3)-generated for any q. In fact, we provide explicit generators x and y of orders 2 and 3, respectively, for the group SL6(q).

The order of every finite group $G$ can be expressed as a product of coprime positive integers ${m}_{1},\cdots ,{m}_{t}$ such that $\pi \left({m}_{i}\right)$ is a connected component of the prime graph of $G$. The integers ${m}_{1},\cdots ,{m}_{t}$ are called the order components of $G$. Some non-abelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symplectic groups ${C}_{2}\left(q\right)$ where $q>5$ are also uniquely determined by their order components. As corollaries of this result, the validities of a...

Let $G$ be a finite group and ${\pi}_{e}\left(G\right)$ be the set of element orders of $G$. Let $k\in {\pi}_{e}\left(G\right)$ and ${m}_{k}$ be the number of elements of order $k$ in $G$. Set $\mathrm{nse}\left(G\right):=\{{m}_{k}:k\in {\pi}_{e}\left(G\right)\}$. In fact $\mathrm{nse}\left(G\right)$ is the set of sizes of elements with the same order in $G$. In this paper, by $\mathrm{nse}\left(G\right)$ and order, we give a new characterization of finite projective special linear groups ${L}_{2}\left(p\right)$ over a field with $p$ elements, where $p$ is prime. We prove the following theorem: If $G$ is a group such that $\left|G\right|=|{L}_{2}\left(p\right)|$ and $\mathrm{nse}\left(G\right)$ consists of $1$, ${p}^{2}-1$, $p(p+\u03f5)/2$ and some numbers divisible by $2p$, where $p$ is a prime greater than...

Let $\omega \left(G\right)$ denote the set of element orders of a finite group $G$. If $H$ is a finite non-abelian simple group and $\omega \left(H\right)=\omega \left(G\right)$ implies $G$ contains a unique non-abelian composition factor isomorphic to $H$, then $G$ is called quasirecognizable by the set of its element orders. In this paper we will prove that the group $PS{L}_{4}\left(5\right)$ is quasirecognizable.

Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $\left|\mathrm{PSL}\right(2,{p}^{2}\left)\right|$ such that $G$ has an irreducible character of degree ${p}^{2}$ and we know that $G$ has no irreducible character $\theta $ such that $2p\mid \theta \left(1\right)$, then $G$ is isomorphic to $\mathrm{PSL}(2,{p}^{2})$. As a consequence of our result we prove that $\mathrm{PSL}(2,{p}^{2})$ is uniquely determined by the structure of its complex group algebra.

Let $G$ be a finite group and $nse\left(G\right)$ the set of numbers of elements with the same order in $G$. In this paper, we prove that a finite group $G$ is isomorphic to $M$, where $M$ is one of the Mathieu groups, if and only if the following hold: (1) $\left|G\right|=\left|M\right|$, (2) $nse\left(G\right)=nse\left(M\right)$.

Let $G$ be a group and $\omega \left(G\right)$ be the set of element orders of $G$. Let $k\in \omega \left(G\right)$ and ${m}_{k}\left(G\right)$ be the number of elements of order $k$ in $G$. Let nse$\left(G\right)=\{{m}_{k}\left(G\right):k\in \omega \left(G\right)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$\left(G\right)=$ nse$\left({S}_{r}\right)$, where ${S}_{r}$ is the symmetric group of degree $r$. In this paper we prove that $G\cong {S}_{r}$, if $r$ divides the order of $G$ and ${r}^{2}$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

$G(3,m,n)$ is the group presented by $\langle a,b\mid {a}^{5}={\left(ab\right)}^{2}={b}^{m+3}{a}^{-n}{b}^{m}{a}^{-n}=1\rangle $. In this paper, we study the structure of $G(3,m,n)$. We also give a new efficient presentation for the Projective Special Linear group $PSL(2,5)$ and in particular we prove that $PSL(2,5)$ is isomorphic to $G(3,m,n)$ under certain conditions.