All finite simple groups of Lie type of rank $n$ over a field of size $q$, with the possible exception of the Ree groups ${}^2{G}_2\left(q\right)$, have presentations with at most 49 relations and bit-length $O(\mathrm{𝚕𝚘𝚐}n+\mathrm{𝚕𝚘𝚐}q)$. Moreover, $A_n$ and $S_n$ have presentations with 3 generators; 7 relations and bit-length $O(\mathrm{𝚕𝚘𝚐}n)$, while $\mathrm{𝚂𝙻}(n,q)$ has a presentation with 6 generators, 25 relations and bit-length $O(\mathrm{𝚕𝚘𝚐}n+\mathrm{𝚕𝚘𝚐}q)$.

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$, then for every subset $B$ of $G$ with $\left|B\right|>\left|G\right|/k^{1/3}$ we have $B^3=G$. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k\ge 2$, then $G$ has a proper subgroup...

Let $G$ be a finite group and let $\pi \left(G\right)=\{p_1,p_2,...,p_k\}$ be the set of prime divisors of $\left|G\right|$ for which $p_1<p_2<\cdots <p_k$. The Gruenberg-Kegel graph of $G$, denoted $GK\left(G\right)$, is defined as follows: its vertex set is $\pi \left(G\right)$ and two different vertices $p_i$ and $p_j$ are adjacent by an edge if and only if $G$ contains an element of order $p_i{p}_j$. The degree of a vertex $p_i$ in $\mathrm{GK}\left(G\right)$ is denoted by $d_G\left(p_i\right)$ and the $k$-tuple $D\left(G\right)=(d_G\left(p_1\right),d_G\left(p_2\right),...,d_G\left(p_k\right))$ is said to be the degree pattern of $G$. Moreover, if $\omega \subseteq \pi \left(G\right)$ is the vertex set of a connected component of $GK\left(G\right)$, then the largest $\omega$-number which divides $\left|G\right|$, is said to be an...

In this paper, we first find the set of orders of all elements in some special linear groups over the binary field. Then, we will prove the characterizability of the special linear group $PSL(13,2)$ using only the set of its element orders.