In this study, the notion of isotopy generalized Bol loop is characterized. A isotope a σ-generalized shown to be σ’-generalized if σ’ fixes its (isotope) identity element where some conjugate σ. and only image isotope’s under right nuclear (where σ). It that can constructed using group subgroup it. conjugacy closed central loop.

Algorithms are constructed which, when an explicit presentation of a finitely generated metabelian group G in the variety si1 is given, produce unitary presentations for the derived subgroup G', the centre Z{G), the Fitting subgroup Fit(G), and the Frattini subgroup <p(G). Additional algorithms of independent interest are developed for commutative algebra which construct the associated set of p...

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) $...

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) $v$...

We give an example of a pure group that does not have the independence property, whose Fitting subgroup is neither nilpotent nor definable andwhose soluble radical is neither soluble nor definable. This answers a question asked by E. Jaligot in May 2013. The Fitting subgroupof a stable group is nilpotent and definable (F.Wagner [11]). More generally, the Fitting subgroup of a group that satisfi...

Little is understood about generalized and non-generalized social anxiety disorder (SAD) and their associations with co-occurring internalizing and externalizing problems in adolescence. In the present study, we investigated adolescent boys with SAD symptoms and considered depressive symptoms as well as antisocial behaviors when looking for patterns during two developmental time periods; junior...

In this paper completely simple semigroups or generalized groups are considered. We characterize generalized groups which are normal generalized groups. Homomorphisms of generalized groups are considered. Equivalent conditions for the kernel of a homomorphism are deduced. We prove that the group components of a generalized group have the same cardinality. We also prove that if G is a finite nor...