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Register a new userOn one generalization of finite nilpotent groups
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Let $sigma ={sigma_{i} | iin I}$ be a partition of the set $Bbb{P}$ of all primes and $G$ a finite group. A chief factor $H/K$ of $G$ is said to be $sigma$-central if the semidirect product $(H/K)rtimes (G/C_{G}(H/K))$ is a $sigma_{i}$-group for some $i=i(H/K)$. $G$ is called $sigma$-nilpotent if every chief factor of $G$ is $sigma$-central. We say that $G$ is semi-${sigma}$-nilpotent (respectively weakly semi-${sigma}$-nilpotent) if the normalizer $N_{G}(A)$ of every non-normal (respectively every non-subnormal) $sigma$-nilpotent subgroup $A$ of $G$ is $sigma$-nilpotent. In this paper we determine the structure of finite semi-${sigma}$-nilpotent and weakly semi-${sigma}$-nilpotent groups.
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It is proved that for any prime $p$ a finitely generated nilpotent group is conjugacy separable in the class of finite $p$-groups if and only if the torsion subgroup of it is a finite $p$-group and the quotient group by the torsion subgroup is abelian.
The description of nilpotent Chernikov $p$-groups with elementary tops is reduced to the study of tuples of skew-symmetric bilinear forms over the residue field $mathbb{F}_p$. If $p e2$ and the bottom of the group only consists of $2$ quasi-cyclic summands, a complete classification is given. We use the technique of quivers with relations.
A theorem of Dolfi, Herzog, Kaplan, and Lev cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in cite{DHKL}, we establish the following generalisation: if $mathfrak{X}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $overline{mathfrak{X}}$ is the extension-closure of $mathfrak{X}$, then there exists an (optimal) constant $gamma$ depending only on $mathfrak{X}$ such that, for all non-trivial finite groups $G$ with trivial $mathfrak{X}$-radical, begin{equation} leftlvert G^{overline{mathfrak{X}}}rightrvert ,>, vert Gvert^gamma, end{equation} where $G^{overline{mathfrak{X}}}$ is the ${overline{mathfrak{X}}}$-residual of $G$. When $mathfrak{X} = mathfrak{N}$, the class of finite nilpotent groups, it follows that $overline{mathfrak{X}} = mathfrak{S}$, the class of finite soluble groups, thus we recover the original theorem of Dolfi, Herzog, Kaplan, and Lev. In the last section of our paper, building on J.,G. Thompsons classification of minimal simple groups, we exhibit a family of subgroup-closed Fitting formations $mathfrak{X}$ of full characteristic such that $mathfrak{S} subset overline{mathfrak{X}} subset mathfrak{E}$, thus providing applications of our main result beyond the reach of cite[Thm.~C]{DHKL}.
Full residual finiteness growth of a finitely generated group $G$ measures how efficiently word metric $n$-balls of $G$ inject into finite quotients of $G$. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of $G$ has finite index in the center of $G$ we show that the growth is precisely $n^b$, where $b$ is the product of the nilpotency class and dimension of $G$. In the general case, we give a method for finding an upper bound of the form $n^b$ where $b$ is a natural number determined by what we call a terraced filtration of $G$. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.
Throughout this paper, all groups are finite. Let $sigma ={sigma_{i} | iin I }$ be some partition of the set of all primes $Bbb{P}$. If $n$ is an integer, the symbol $sigma (n)$ denotes the set ${sigma_{i} |sigma_{i}cap pi (n) e emptyset }$. The integers $n$ and $m$ are called $sigma$-coprime if $sigma (n)cap sigma (m)=emptyset$. Let $t > 1$ be a natural number and let $mathfrak{F}$ be a class of groups. Then we say that $mathfrak{F}$ is $Sigma_{t}^{sigma}$-closed provided $mathfrak{F}$ contains each group $G$ with subgroups $A_{1}, ldots , A_{t}in mathfrak{F}$ whose indices $|G:A_{1}|$, $ldots$, $|G:A_{t}|$ are pairwise $sigma$-coprime. In this paper, we study $Sigma_{t}^{sigma}$-closed classes of finite groups.
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