Related papers: A criterion for a normal subgroup to be hypercentr…

Let G be a finite group and H a normal subgroup such that G/H is cyclic. Given a conjugacy class g^G of G we define its centralizing subgroup to be HC_G(g). Let K be such that H\le K\le G. We show that the G-conjugacy classes contained in K…

Let $\sigma =\{\sigma_{i} | i\in I\}$ 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…

Let G be a group and H be a subgroup of G which is either finite or of finite index in G. In this note, we give some characterizations for normality of H in G. As a consequence we get a very short and elementary proof of the Main Theorem of…

A p-group G is p-central if the central quotient has exponent p. We prove that for a subset of finite p-central p-groups, the order of the group G divides the order of Aut(G).

We initiate the study of profinite groups of non-negative deficiency. The principal focus of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group $G$ of non-negative…

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.

Let $G$ be a word hyperbolic group in the sense of Gromov and $P$ its associated Rips complex. We prove that the fixed point set $P^H$ is contractible for every finite subgroups $H$ of $G$. This is the main ingredient for proving that $P$…

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is…

We examine $p$-groups with the property that every non-normal subgroup has a normalizer which is a maximal subgroup. In particular we show that for such a $p$-group $G$, when $p=2$, the center of $G$ has index at most 16 and when $p$ is odd…

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…

We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite $p$-groups and finite simple groups with…

The greatest power of a prime $p$ dividing the natural number $n$ will be denoted by $n_p$. Let $Ind_G(g)=|G:C_G(g)|$. Suppose that $G$ is a finite group and $p$ is a prime. We prove that if there exists an integer $\alpha>0$ such that…

We study the pro-$p$ group $G$ whose finite quotients give the prototypical Sylow $p$-subgroup of the general linear groups over a finite field of prime characteristic $p$. In this article, we extend the known results on the subgroup…

Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is…

Gcharacter tables of a finite group G were defined before. These tables can be very useful to obtain certain structural information of a normal subgroup from the character table of G. We analyze certain structural properties of normal…

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the $ \Pi $-property in $ G $ if for any chief factor $ L / K $ of $ G $, $ |G/K : N_{G/K}(HK/K\cap L/K )| $ is a $ \pi (HK/K\cap L/K) $-number. In this paper, we…

Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group of…

We prove that the isomorphism type of the subrack lattice of a finite group determines the nilpotence class. We analyze the problem of estimating the orders of the group elements corresponding to the atoms of the subrack lattice. As a…

We prove the conjugacy of Sylow $p$-subgroups of linear pseudofinite groups under the assumption of the existence of a finite Sylow $p$-subgroup. We also give an example of a linear pseudofinite group with non-conjugate Sylow $2$-subgroups.

Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this note we determine all the large maximal subgroups of finite simple groups, and we establish an…