let l = u3(9) be the simple projective unitary group in dimension 3 over a eld with 92 elements. in this article, we classify groups with the same order and degree pattern as an almost simple group related to l. since aut(l) = z4 hence almost simple groups related to l are l, l : 2 or l : 4. in fact, we prove that l, l : 2 and l : 4 are od-characterizable.

a $p$-group $g$ is called a $mathcal{cac}$-$p$-group if $c_g(h)/h$ is ‎cyclic for every non-cyclic abelian subgroup $h$ in $g$ with $hnleq‎ ‎z(g)$‎. ‎in this paper‎, ‎we give a complete classification of‎ ‎finite $mathcal{cac}$-$p$-groups‎.

let g be a group. a subset x of g is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in x. if |x| ≥ |y | for any other set of pairwise non-commuting elements y in g, then x is said to be a maximal subset of pairwise non-commuting elements. in this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...

let $g$ be a finite group‎. ‎we denote by $psi(g)$ the integer $sum_{gin g}o(g)$‎, ‎where $o(g)$ denotes the order of $g in g$‎. ‎here we show that‎ ‎$psi(a_5)< psi(g)$ for every non-simple group $g$ of order $60$‎, ‎where $a_5$ is the alternating group of degree $5$‎. ‎also we prove that $psi(psl(2,7))

We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every finite group F is obtained as a quotient of an open subgroup of G. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann....

The main theorem is that if G is a pseudofinite group with stable theory, then G has a definable normal soluble subgroup of finite index.