### $1{\textstyle \frac{1}{2}}$-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) $nse\left(G\right)=nse\left(M\right)$.

One of the important questions that remains after the classification of the finite simple groups is how to recognize a simple group via specific properties. For example, authors have been able to use graphs associated to element orders and to number of elements with specific orders to determine simple groups up to isomorphism. In this paper, we prove that Suzuki groups $Sz\left(q\right)$, where $q\pm \sqrt{2q}+1$ is a prime number can be uniquely determined by the order of group and the number of elements with the same order.

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.