Let $G(\circ )$ and $G(*)$ be two groups of finite order $n$, and suppose that they share a normal subgroup $S$ such that $u\circ v=u*v$ if $u\in S$ or $v\in S$. Cases when $G/S$ is cyclic or dihedral and when $u\circ v\ne u*v$ for exactly $n^2/4$ pairs $(u,v)\in G\times G$ have been shown to be of crucial importance when studying pairs of 2-groups with the latter property. In such cases one can describe two general constructions how to get all possible $G(*)$ from a given $G=G(\circ )$. The constructions, denoted by $G[\alpha ,h]$ and $G[\beta ,\gamma ,h]$, respectively, depend on a coset $\alpha$ (or two cosets $\beta$ and $\gamma$) modulo $S$, and on an...

A group $G$ has the endomorphism kernel property (EKP) if every congruence relation $\theta$ on $G$ is the kernel of an endomorphism on $G$. In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP.

In the first part of this paper we prove without using the transfer or characters the equivalence of some conditions, each of which would imply p-nilpotence of a finite group G. The implication of p-nilpotence also can be deduced without the transfer or characters if the group is p-constrained. For p-constrained groups we also prove an equivalent condition so that $O^{q^{\text{'}}}\left(G\right)P$ should be p-nilpotent. We show an example that this result is not true for some non-p-constrained groups. In the second part of the...

A group $G$ has all of its subgroups normal-by-finite if $H/H_G$ is finite for all subgroups $H$ of $G$. The Tarski-groups provide examples of $p$-groups ($p$ a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a $2$-group with every subgroup normal-by-finite is locally finite. We also prove that if $|H/H_G|\le 2$ for every subgroup $H$ of $G$, then $G$ contains an Abelian subgroup of index at most $8$.

In this paper, we consider finite groups with precisely one nonlinear nonfaithful irreducible character. We show that the groups of order 16 with nilpotency class 3 are the only $p$-groups with this property. Moreover we completely characterize the nilpotent groups with this property. Also we show that if $G$ is a group with a nontrivial center which possesses precisely one nonlinear nonfaithful irreducible character then $G$ is solvable.

In this paper it is proved that a finite group G with an automorphism $\alpha$ of prime order r, such that $C_G\left(\alpha \right)=1$ is contained in a nilpotent subgroup H, with $\left(\left|H\right|,r\right)=1$, is nilpotent provided that either $\left|H\right|$ is odd or, if $\left|H\right|$ is even, then r is not a Fermât prime.

In 1954, Kontorovich and Plotkin introduced the concept of a modular chain in a lattice to obtain a lattice-theoretic characterization of the class of torsion-free nilpotent groups. We determine the structure of finite groups with modular chains. It turns out that this class of groups lies strictly between the class of finite groups with lower semimodular subgroup lattice and the projective closure of the class of finite nilpotent groups.

Let $G$ be a finite group, $p$ the smallest prime dividing the order of $G$ and $P$ a Sylow $p$-subgroup of $G$ with the smallest generator number $d$. There is a set ${ℳ}_d\left(P\right)=\{P_1,P_2,\cdots ,P_d\}$ of maximal subgroups of $P$ such that ${\bigcap }_{i=1}^d{P}_i=\Phi \left(P\right)$. In the present paper, we investigate the structure of a finite group under the assumption that every member of ${ℳ}_d\left(P\right)$ is either $s$-permutably embedded or weakly $s$-permutable in $G$ to give criteria for a group to be $p$-supersolvable or $p$-nilpotent.