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...

here we consider all finite non-abelian 2-generator $p$-groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$-roots of them. also we find integers $n$ for which, these groups are $n$-central.

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).

in this paper, we consider the finitely presented groups $g_{m}$ and $k(s,l)$ as follows;$$g_{m}=langle a,b| a^m=b^m=1,~[a,b]^a=[a,b],~[a,b]^b=[a,b]rangle $$$$k(s,l)=langle a,b|ab^s=b^la,~ba^s=a^lbrangle;$$and find the $n^{th}$-commutativity degree for each of them. also we study the concept of $n$-abelianity on these groups, where $m,n,s$ and $l$ are positive integers, $m,ngeq 2$ and $g.c.d(s,...

The probability that the commutator of two group elements is equal to a given element has been introduced in literature few years ago. Several authors have investigated this notion with methods of the representation theory and with combinatorial techniques. Here we illustrate that a wider context may be considered and show some structural restrictions on the 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.

We study Kuzmin’s conjecture on the index of nilpotency for the variety N il5 of associative nil-algebras of degree 5. Due to Vaughan-Lee [11] the problem is reduced to that for k-generator N il5-superalgebras, where k ≤ 5. We confirm Kuzmin’s conjecture for 2-generator superalgebras proving that they are nilpotent of degree 15.

the degree kirchhoff index of a connected graph $g$ is defined as‎ ‎the sum of the terms $d_i,d_j,r_{ij}$ over all pairs of vertices‎, ‎where $d_i$ is the‎ ‎degree of the $i$-th vertex‎, ‎and $r_{ij}$ the resistance distance between the $i$-th and‎ ‎$j$-th vertex of $g$‎. ‎bounds for the degree kirchhoff index of the line and para-line‎ ‎graphs are determined‎. ‎the special case of regular grap...

One of the most interesting aspects in the theory of cellular automata is the study of different types of nilpotency, that is, different ways in which a cellular automaton can force a particular symbol (usually called 0) to appear frequently in all its spacetime diagrams. The simplest such notion, called simply ‘nilpotency’, is that the cellular automaton c maps every configuration to a uniform...