the divisibility graph $mathscr{d}(g)$ for a finite group $g$ is a graph with vertex set ${rm cs}(g)setminus{1}$‎ ‎where ${rm cs}(g)$ is the set of conjugacy class sizes of $g$‎. ‎two vertices $a$ and $b$ are adjacent whenever $a$ divides‎ ‎$b$ or $b$ divides $a$‎. ‎in this paper we will find the number of connected components of $mathscr{d}(g)$ where $g$ is a‎ ‎simple zassenhaus group or an sp...

‎there is much interest in finding short presentations for the finite‎ ‎simple groups‎. ‎indeed it has been suggested that all these groups are‎ ‎efficient in a technical sense‎. ‎in previous papers we produced nice‎ ‎efficient presentations for all except one of the simple groups with‎ ‎order less than one million‎. ‎here we show that all simple groups with‎ ‎order between $1$ million and $5$ ...

let $g$ be a finite group and let $text{cd}(g)$ be the set of all‎ ‎complex irreducible character degrees of $g$‎. ‎b‎. ‎huppert conjectured‎ ‎that if $h$ is a finite nonabelian simple group such that‎ ‎$text{cd}(g) =text{cd}(h)$‎, ‎then $gcong h times a$‎, ‎where $a$ is‎ ‎an abelian group‎. ‎in this paper‎, ‎we verify the conjecture for‎ ${f_4(2)}.$‎

The spectrum of a finite group is the set element orders this group. main goal paper to survey results concerning recognition simple groups by spectrum, in particular, list all for which problem solved.

in this paper, it was shown that , where  and , and , where  is not prime and , are od-characterizable.

‎let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $f$ of characteristic $pgeq 0$‎. ‎we survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2‎, ‎3‎, ‎7)$-generated and of those which are $(2,3)$-generated‎.

‎let us consider the set of non-abelian finite simple groups which‎ ‎admit non-trivial irreducible projective representations of degree $le 7$ over‎ ‎an algebraically closed field $mathbb{f}$ of characteristic $pgeq 0$‎. ‎we survey some recent results which‎ ‎lead to the complete list of the groups in this set which are‎ ‎$(2,3,7)$-generated and of those which are $(2,3)$-generated‎.