The paper is concerned with the class of groups satisfying the finite embedding (FE) property. This is a generalization of residually finite groups. In [2] it was asked whether there exist FE-groups which are not residually finite. Here we present such examples. To do this, we construct a family of three-generator soluble FE-groups with torsion-free abelian factors. We study necessary and sufficient conditions for groups from this class to be residually finite. This answers the questions asked in...

Let $ℱ$ be a saturated formation containing the class of supersolvable groups and let $G$ be a finite group. The following theorems are presented: (1) $G\in ℱ$ if and only if there is a normal subgroup $H$ such that $G/H\in ℱ$ and every maximal subgroup of all Sylow subgroups of $H$ is either $c$-normal or $S$-quasinormally embedded in $G$. (2) $G\in ℱ$ if and only if there is a normal subgroup $H$ such that $G/H\in ℱ$ and every maximal subgroup of all Sylow subgroups of $F^{*}\left(H\right)$, the generalized Fitting subgroup of $H$, is either $c$-normal or $S$-quasinormally...

In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group $\mathbf{G}$ defined over a finite field with corresponding Frobenius map $F$ and derive the number of $F$-stable semisimple classes of $\mathbf{G}$ with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters of the fixed point subgroup ${\mathbf{G}}^F$ to their inertia group in the full automorphism group. As a consequence, we...

Let $$A=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right],B_{\lambda }=\left[\begin{array}{cc}1& 0\\ \lambda & 1\end{array}\right].$$ We call a complex number $\lambda$ “semigroup free“ if the semigroup generated by $A$ and $B_{\lambda }$ is free and “free” if the group generated by $A$ and $B_{\lambda }$ is free. First families of semigroup free $\lambda$’s were described by J. L. Brenner, A. Charnow (1978). In this paper we enlarge the set of known semigroup free $\lambda$’s. To do it, we use a new version of “Ping-Pong Lemma” for semigroups embeddable in groups. At the end we present most of the known results related to semigroup free and free numbers in a common picture....

Let G be some metabelian 2-group satisfying the condition G/G’ ≃ ℤ/2ℤ × ℤ/2ℤ × ℤ/2ℤ. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem for the 2-ideal classes of some fields k satisfying the condition $Gal(k{₂}^{\left(2\right)}/k)\simeq G$, where $k{₂}^{\left(2\right)}$ is the second Hilbert 2-class field of k.

New results on tight connections among pronormal, abnormal and contranormal subgroups of a group have been established. In particular, new characteristics of pronormal and abnormal subgroups have been obtained.

The article is dedicated to groups in which the set of abnormal and normal subgroups ($U$-subgroups) forms a lattice. A complete description of these groups under the additional restriction that every counternormal subgroup is abnormal is obtained.

A recent result of Balandraud shows that for every subset $S$ of an abelian group $G$ there exists a non trivial subgroup $H$ such that $\left|TS\right|\le \left|T\right|+\left|S\right|-2$ holds only if $H\subset Stab\left(TS\right)$. Notice that Kneser’s Theorem only gives $\left\{1\right\}\ne Stab\left(TS\right)$.This strong form of Kneser’s theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabelian groups and, by using classical tools from Additive Number Theory, extend some of the above results. In particular we obtain short proofs of Balandraud’s...