Related papers: Powerfully nilpotent groups

For any group G, let C(G) denote the intersection of the normal- izers of centralizers of all elements of G. Set C0 = 1. Define Ci+1(G)=Ci(G) = C(G=Ci(G)) for i ? 0. By C1(G) denote the terminal term of the ascending series. In this paper,…

The enhanced power graph of a group is the simple graph whose vertex set is consisted of all elements of the group, and whose any pair of vertices are adjacent if they generate a cyclic subgroup. In this paper, we classify all finite groups…

Let $k(G)$ be the number of conjugacy classes of finite groups $G$ and $\pi_e(G)$ be the set of the orders of elements in $G$. Then there exists a non-negative integer $k$ such that $k(G)=|\pi_e(G)|+k$. We call such groups to be $co(k)$…

Let p be a fixed prime. An Abelian p-group is an Abelian group (not necessarily finitely generated) in which every element has for its order some power of p. The countable Abelian p-groups are classified by Ulm's theorem, and Khisamiev…

The pro-isomorphic zeta function of a finitely generated nilpotent group $\Gamma$ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of $\Gamma$. Such zeta functions…

In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every…

A group is said to be strongly amenable if each of its proximal topological actions has a fixed point. We show that a finitely generated group is strongly amenable if and only if it is virtually nilpotent. More generally, a countable…

Let $G$ be a non-abelian $p$-group of order $p^n$ and $M(G)$ denote the Schur multiplier of $G$. Niroomand proved that $|M(G)| \leq p^{\frac{1}{2}(n+k-2)(n-k-1)+1}$ for non-abelian $p$-groups $G$ of order $p^n$ with derived subgroup of…

A pro-p Cappitt group is a pro-p group G such that the subgroup topologically generated by all non-normal closed subgroups is a proper subgroup of G. In this paper we prove that non-abelian pro-p Cappitt groups whose torsion subgroup is…

Let p be a prime number and let c, d be natural numbers. Then, the number of possible isomorphism types for the mod p cohomology algebra of a d-generated finite p-group of nilpotency class c is bounded by a function depending only on p, c…

Let G be a finite group of order n and V an irreducible representation over the complex numbers of dimension d. For some nonnegative number e, we have n=d(d+e). If e is small, then the character of V has unusually large degree. We fix e and…

We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as…

We explore some of the special features with respect to Bredon cohomology of groups having all its finite subgroups either nilpotent or p-groups or cyclic p-groups. We get some results on dimensions and also a formula for the equivariant…

In this article we develop the theory of residually finite rationally $p$ (RFR$p$) groups, where $p$ is a prime. We first prove a series of results about the structure of finitely generated RFR$p$ groups (either for a single prime $p$, or…

We prove that a finitely generated Lie algebra $L$ such that (i) every commutator in generators is ad-nilpotent, and (ii) $ L$ satisfies a polynomial identity, is nilpotent. As a corollary we get that a finitely generated residually-$p$…

Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities…

We develop general formulae for the numbers of conjugacy classes and irreducible complex characters of finite p-groups of nilpotency class less than p. This allows us to unify and generalize a number of existing enumerative results, and to…

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to…

The authors have shown previously that every locally pro-p contraction group decomposes into the direct product of a p-adic analytic factor and a torsion factor. It has long been known that p-adic analytic contraction groups are nilpotent.…

We prove that every free metabelian non--cyclic group has a finitely generated isolated subgroup which is not separable in the class of nilpotent groups. As a corollary we prove that for every prime number $p$ an arbitrary free metabelian…