Related papers: On an extension of Shlyk's theorem

For a group $G$ and a subgroup $H$ of $G$ this article discusses the normalizer of $H$ in the units of a group ring $RG$. We prove that $H$ is only normalized by the `obvious' units, namely products of elements of $G$ normalizing $H$ and…

Let G be a finite non-nilpotent group such that every Sylow subgroup of G is generated by at most d elements, and such that p is the largest prime dividing |G|. We show that G has a non-nilpotent image G/N, such that N is characteristic and…

In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.

In this paper, we investigate the structure of finite group G by assuming that the intersections between p-sylowizers of some p-subgroups of G and $O^p(G)$ are S-permutable in G. We obtain some criterions for p-nilpotency of a finite group.

If G is a non-nilpotent group and nil(G) = {g \in G : <g, h> is nilpotent for all h\in G}, the nilpotent graph of G is the graph with set of vertices G-nil(G) in which two distinct vertices are related if they generate a nilpotent subgroup…

In this paper we study the groups all whose maximal or all Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product the with generalizations of the Fitting subgroup $\mathrm{F}^*(G)$ and $\mathrm{\tilde F}(G)$. We prove that a…

We investigate the intersection of normalizers and $\mathfrak{F}$-subnormalizers of different types of systems of subgroups ($\mathfrak{F}$-maximal, Sylow, cyclic primary). We described all formations $\mathfrak{F}=\underset{i\in…

We show that every finite group $G$ of size at least $3$ has a nilpotent subgroup of class at most $2$ and size at least $|G|^{1/32\log\log|G|}$. This answers a question of Pyber, and is essentially best possible.

A maximal abelian normal subgroup A in a nilpotent group N is self-centralizing. This makes their role an important one in determining the structure of the nilpotent group. For example if A is finite then N is also finite. In the free…

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…

This is a long overdue write up of the following: If the fundamental group of a normal complex algebraic variety (respectively Zariski open subset of a compact K\"ahler manifold) is a solvable group of matrices over Q (respectively…

Every finite non-nilpotent group can be extended by a term operation such that solving equations in the resulting algebra is NP-complete and checking identities is co-NP-complete. This result was firstly proven by Horv\'ath and Szab\'o; the…

Let $G$ be a finite group and let $\mathfrak{F}$ be a hereditary saturated formation. We denote by $\mathbf{Z}_{\mathfrak{F}}(G)$ the product of all normal subgroups $N$ of $G$ such that every chief factor $H/K$ of $G$ below $N$ is…

A group $G$ is said to be totally $k$-closed for a positive integer $k$ if, in each of its faithful permutation representations on a set $\Omega^k$, $G$ is the largest subgroup of the symmetric group $\operatorname{Sym}(\Omega)$ that…

In this paper, we establish the theory of nilpotent hypergroups and study some properties of nilpotent hypergroups and provided some structural characterizations of nilpotent hypergroups.

A subgroup $H$ is said to be $nc$-supplemented in a group $G$ if there is a subgroup $K\leq G$ such that $HK\unlhd G$ and $H\cap K$ is contained in $H_G$, the core of $H$ in $G$. We characterize the solvability of finite groups $G$ with…

We give a new proof of Gromov's theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally…

We prove that the symmetric group $S_n$ has a unique minimal cover $\mathcal{M}$ by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the order of $\mathcal{M}$. In addition, we prove that the order of…

Viewing Kan complexes as $\infty$-groupoids implies that pointed and connected Kan complexes are to be viewed as $\infty$-groups. A fundamental question is then: to what extent can one "do group theory" with these objects? In this paper we…

Let $G$ be a non-abelian finite simple group. A famous result of Liebeck and Shalev is that there is an absolute constant $c$ such that whenever $S$ is a non-trivial normal subset in $G$ then $S^{k} = G$ for any integer $k$ at least $c…