By T. Kepka and M. Niemenmaa if the inner mapping group of a finite loop Q is abelian, then the loop Q is centrally nilpotent. For a long time there was no example of a nilpotency degree greater than two. In the nineties T. Kepka raised the following problem: whether every finite loop with abelian inner mapping group is centrally nilpotent of class at most two. For many years the prevailing opi...

Suppose n is a fixed positive integer. We introduce the relative n-th non-commuting graph ΓH,G, associated to the nonabelian subgroup H of group G. The vertex set is G \ C H,G in which C H,G = {x ∈ G : [x, y] = 1 and [x, y] = 1 for all y ∈ H}. Moreover, {x, y} is an edge if x or y belong to H and xy 6= yx or xy 6= yx. In fact, the relative n-th commutativity degree, Pn(H,G) the probability that...

We prove a conjecture made by Brundan and Kleshchev on the nilpotency degree of cyclotomic quotients of rings that categorify one-half of quantum sl(k).

let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

We present some results on character degree sums in connection with several important characteristics of finite groups such as p-solvability, solvability, supersolvability, and nilpotency. Some of them strengthen known results in the literature.