We shall say that an automorphism a is nilpotent or acts nilpotently on a group G if in the holomorph H= [G](a) of G with a, a is a bounded left Engel element, that is, [H, ¿a] = l for some natural number ¿. Here [H, ka] means [H, (k — l)a] with [H, Oa] denoting H. Let G' denote the commutator subgroup [G, G], and let $(G) denote the Frattini subgroup of G. If a is an automorphism of a nilpoten...

If K is a closed subgroup of compact group G, the probability that randomly chosen pair elements from and G commute denoted by Pr(K, G). Say ≤ ϵ-central in if Pr(〈g〉, G) ≥ ϵ for any g K. Here 〈g〉 denotes monothetic generated ∈ G. Our main result then there an ϵ-bounded number e normal T such both index [G:T] order commutator [Ke, T] are finite ϵ-bounded. In particular, which > 0 [Ge,

In groups, an abelian normal subgroup induces congruence. We construct a class of centrally nilpotent Moufang loops containing subloop that does not induce On the other hand, we prove in 6-divisible loops, every congruence solvability adopted from universal-algebraic commutator theory modular varieties is strictly stronger than classical group theory. It open problem whether two notions coincid...

Let $K$ be a subgroup of finite group $G$ . The probability that an element commutes with is denoted by $Pr(K,G)$ Assume $Pr(K,G)\geq \epsilon$ for some fixed $\epsilon >0$ We show there normal $T\leq G$ and $B\leq K$ such the indices $[G:T]$ $[K:B]$ order commutator $[T,B]$ are $\epsilon$ -bounded. This extends well-known theorem, due to P. M. Neumann, covers case where $K=G$ deduce number ...

We call a finite group metabelian if its commutator subgroup is abelian. Thus a split metabelian group is a split extension of one abelian group by a second (possibly trivial) abelian group. In this paper we devise an algorithm for the construction of the character table of any split metabelian group G. The only requisites are that the reader be capable of deriving the character table for an ab...

The residual torsion-free nilpotence of the commutator subgroup a knot group has played key role in studying bi-orderability groups. A technique developed by Mayland provides sufficient condition for to be residually-torsion-free nilpotent using work Baumslag. In this paper, we apply Mayland's several genus one pretzel knots and family with arbitrarily high genus. As result, obtain large number...

‎let $g$ be a finite group‎. ‎an element $gin g$ is called non-vanishing‎, ‎if for‎ ‎every irreducible complex character $chi$ of $g$‎, ‎$chi(g)neq 0$‎. ‎the bi-cayley graph $bcay(g,t)$ of $g$ with respect to a subset $tsubseteq g$‎, ‎is an undirected graph with‎ ‎vertex set $gtimes{1,2}$ and edge set ${{(x,1),(tx,2)}mid xin g‎, ‎ tin t}$‎. ‎let $nv(g)$ be the set‎ ‎of all non-vanishing element...

Abstract We introduce and study families of finite index subgroups the modular group that generalize congruence subgroups. Such groups, termed ϕ‐congruence subgroups, are obtained by reducing homomorphisms ϕ from into a linear algebraic modulo integers. In particular, we examine two examples, arising on one hand map quasi‐unipotent group, other maps symplectic groups degree four. case, also pro...

Szasz [l ] has recently shown that a group is cyclic if and only if it satisfies condition (A) below. (A) Every cyclic subgroup of the group is for some positive integer k the subgroup generated by the £th powers of the elements of the group. We shall extend this idea here to show that a metacyclic group whose commutator subgroup has order relatively prime to its index is characterized as a sol...

Abstract It is known that if G a group such the centre factor $$G/\zeta (G)$$ G / ζ ( ) polycyclic, then also commutator subgroup $$G'$$ ′ polycyclic. The aim of this paper to des...