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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.
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 su
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 inequ
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
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 ${sigma_{i} |sigma_{i}cap pi (n) e emptyset }$. Th