‎let $s$ be a subset of a finite group $g$‎. ‎the bi-cayley graph ${rm bcay}(g,s)$ of $g$ with respect to $s$ is an undirected graph with vertex set $gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid xin g‎, ‎ sin s}$‎. ‎a bi-cayley graph ${rm bcay}(g,s)$ is called a bci-graph if for any bi-cayley graph ${rm bcay}(g,t)$‎, ‎whenever ${rm bcay}(g,s)cong {rm bcay}(g,t)$ we have $t=gs^alpha$ for some $...

‎we characterize those groups $g$ and vector spaces $v$ such that $v$ is a faithful irreducible $g$-module and such that each $v$ in $v$ is centralized by a $g$-conjugate of a fixed non-identity element of the fitting subgroup $f(g)$ of $g$‎. ‎we also determine those $v$ and $g$ for which $v$ is a faithful quasi-primitive $g$-module and $f(g)$ has no regular orbit‎. ‎we do use these to show in ...

In the past two decades, there have been far-reaching developments in the problem of determining all finite non-abelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been announced. Of course, the solution will have a considerable effect on many related areas, both wi...

We answer in the negative a question of Hartley about representations finite groups, by constructing examples simple groups with arbitrarily large whose endomorphism ring consists just scalars. show as consequence that there are automorphisms locally group SL(∞,Fq) trivial centralizer. The smallest our is A6 q=9.

let $g$ be a finite group‎. ‎a subset $x$ of $g$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $x$ do not commute‎. ‎in this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.

let h be a subgroup of a group g. h is said to be s-embedded in g if g has a normal t such that ht is an s-permutable subgroup of g and h ∩ t ≤ h sg, where h denotes the subgroup generated by all those subgroups of h which are s-permutable in g. in this paper, we investigate the influence of minimal s-embedded subgroups on the structure of finite groups. we determine the structure the finite grou...