the decomposition $gamma=bh$ of a group $gamma$ into a subset $b$ and a subgroup $h$ of $gamma$ induces, under general conditions, a group-like structure for $b$, known as a gyrogroup. the famous concrete realization of a gyrogroup, which motivated the emergence of gyrogroups into the mainstream, is the space of all relativistically admissible velocities along with a binary operation given by t...

let $h$, $l$ and $x$ be subgroups of a finite group$g$. then $h$ is said to be $x$-permutable with $l$ if for some$xin x$ we have $al^{x}=l^{x}a$. we say that $h$ is emph{$x$-quasipermutable } (emph{$x_{s}$-quasipermutable}, respectively) in $g$ provided $g$ has a subgroup$b$ such that $g=n_{g}(h)b$ and $h$ $x$-permutes with $b$ and with all subgroups (with all sylowsubgroups, respectively) $v$...

in this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. with this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. n(g) and s(g) are the set of all nilpotency classes and the set of all solvability classes for the group g with respect to different automorphi...

we consider the class $mathfrak m$ of $bf r$--modules where $bf r$ is an associative ring. let $a$ be a module over a group ring $bf r$$g$, $g$ be a group and let $mathfrak l(g)$ be the set of all proper subgroups of $g$. we suppose that if $h in mathfrak l(g)$ then $a/c_{a}(h)$ belongs to $mathfrak m$. we investigate an $bf r$$g$--module $a$ such that $g not = g'$, $c_{g}(a) = 1$. we stud...

Bois noir is an important grapevine yellows disease in Europe that can cause serious economic losses in grapevine production. It is caused by stolbur phytoplasma strains of the taxonomic group 16SrXII-A. Hyalesthes obsoletus Signoret (Hemiptera: Cixiidae) is the most important vector of bois noir in Europe. This polyphagous planthopper is assumed to mainly use stinging nettle [Urtica dioica L. ...

a finite group g is said to be a pos-group if for each x in g the cardinality of the set {y in g | o(y) = o(x)} is a divisor of the order of g. in this paper we study the structure of pos-groups with some cyclic sylow subgroups.

in this paper we consider the group algebra $r(c_2times‎ ‎d_infty)$‎. ‎it is shown that $r(c_2times d_infty)$ can be‎ ‎represented by a $4times 4$ block circulant matrix‎. ‎it is also‎ ‎shown that $mathcal{u}(mathbb{z}_2(c_2times d_infty))$ is‎ ‎infinitely generated‎.

in this paper, we show how certain metabelian groups can be found within polynomial evaluation groupoids. we show that every finite abelian group can beobtained as a polynomial evaluation groupoid.