International Journal of Group Theory
Volume:7 Issue: 4, Dec 2018

Let $G$ be a groupý, ýwe say that $G$ satisfies the property $\mathcal{T}(\infty)$ provided thatý, ýevery infinite set of elements of $G$ contains elements $x\neq yý, ýz$ such that $[xý, ýyý, ýz]=1=[yý, ýzý, ýx]=[zý, ýxý, ýy]$ý.
ýWe denote by $\mathcal{C}$ the class of all polycyclic groupsý, ý$\mathcal{S}$ the class of all soluble groupsý, ý$\mathcal{R}$ the class of all residually finite groupsý, ý$\mathcal{L}$ the class of all locally graded groupsý, ý$\mathcal{N}_2$ the class of all nilpotent group of class at most twoý, ýand $\mathcal{F}$ the class of all finite groupsý. ýIn this paperý, ýfirst we shall prove that if $G$ is a finitely generated locally graded groupý, ýthen $G$ satisfies $\mathcal{T}(\infty)$ if and only if $G/Z_2(G)$ is finiteý, ýand then we shall conclude that if $G$ is a finitely generated group in $\mathcal{T}(\infty)$ý, ýthený ý\[G\in\mathcal{L}\Leftrightarrow G\in\mathcal{R}\Leftrightarrow G\in\mathcal{S}\Leftrightarrow G\in\mathcal{C}\Leftrightarrow G\in\mathcal{N}_2\mathcal{F}.\]ý

Let $G$ be a group and $Aut^{\Phi}(G)$ denote the group of all automorphisms of $G$ centralizing $G/\Phi(G)$ elementwiseý. ýIn this paperý, ýwe characterize the finite $p$-groups $G$ with cyclic Frattini subgroup for which $|Aut^{\Phi}(G):Inn(G)|=p$ý.

ýThe Magnus embedding of a free metabelian group induces the embedding of partially commutative metabelian group $S_\Gamma$ in a group of matrices $M_\Gamma$. Properties and the universal theory of the group $M_\Gamma$ are studied.

In this paper we investigate the special automata over finite rank free groups and estimate asymptotic characteristics of sets they acceptý. ýWe show how one can decompose an arbitrary regular subset of a finite rank free group into disjoint union of sets accepted by special automata or special monoidsý. ýThese automata allow us to compute explicitly generating functionsý, ý$\lambda-$measures and Cesaro measure of thick monoidsý. ýAlso we improve the asymptotic classification of regular subsets in free groupsý.

ýýIn this paper we prove that the Maschke property holds for coprime actions on some important classes of $p$-groups likeý: ýmetacyclic $p$-groupsý, ý$p$-groups of $p$-rank two for $p>3$ and some weaker property holds in the case of regular $p$-groupsý. ýThe main focus will be the case of coprime actions on the iterated wreath product $P_n$ of cyclic groups of order $p$ý, ýi.eý. ýon Sylow $p$-subgroups of the symmetric groups $S_{p^n}$ý, ýwhere we also prove that a stronger form of the Maschke property holdsý. ýThese results contribute to a future possible classification of all $p$-groups with the Maschke propertyý. ýWe apply these results to describe which normal partition subgroups of $P_n$ have a complementý. ýIn the end we also describe abelian subgroups of $P_n$ of largest sizeý.