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.

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

In this paper, we introduce and study the notion of precompacttopological generalized groups and some new results are given.

Every Seifert Fibre Group is the lift of a Fuchsian group to the universal covering group of PSL (2,R). From this, we work out a form of presentation for such a group. With the calculation of the Euler number, we can establish the presentation of the derived group of a Seifert Fibre Group

a famous theorem of schur states that for a group $g$ finiteness of $g/z(g)$‎ ‎implies the finiteness of $g'.$ the converse of schur's theorem is an interesting problem which has been considered by some‎ ‎authors‎. ‎recently‎, ‎podoski and szegedy proved the truth of the converse of schur's theorem for capable groups‎. ‎they also established an explicit bound for the index of the center of such...

in this paper we find systems of subgroups of a finite‎ ‎group‎, ‎which $bbb p$nobreakdash-hspace{0pt}subnormality guarantees supersolvability‎ ‎of the whole group‎.

let $g$ be a non-abelian group of order $p^n$‎, ‎where $nleq 5$ in which $g$ is not extra special of order $p^5$‎. ‎in this paper we determine the maximal size of subsets $x$ of $g$‎ ‎with the property that $xyneq yx$ for any $x,y$ in $x$ with‎ ‎$xneq y$‎.