Let $G$ be a group and $Aut(G)$ be the group of automorphisms of‎ ‎$G$‎. ‎For any natural‎ number $m$‎, ‎the $m^{th}$-autocommutator subgroup of $G$ is defined‎ ‎as‎: ‎$$K_{m} (G)=langle[g,alpha_{1},ldots,alpha_{m}] |gin G‎,‎alpha_{1},ldots,alpha_{m}in Aut(G)rangle.$$‎ ‎In this paper‎, ‎we obtain the $m^{th}$-autocommutator subgroup of‎ ‎all finite abelian groups‎.

specific elements (cr, al, ca, and si) and loss on ignition (loi) were used for semi-quantitative determination of the normative mineralogical composition of naturally occurring chromium ores in ophiolite complexes. by far the occurring ore minerals are complex both from the mineralogical and chemical viewpoints. the secondary minerals serpentine and cr-containing chlorite (kaemmererite) form n...

The early interest in selenium (Se) focused on its toxicity, but since 1957 it has been recognized as an essential dietary element. The diet is the major Se source and approximately 80% of dietary Se is absorbed, depending on the type of food consumed. In several regions of the world, the content of Se in diets has been estimated as insufficient. The dietary requirement for Se for most species ...

‎In this study‎, ‎Frattini supplement subgroup and Frattini supplemented group‎ ‎are defined by Frattini subgroup‎. ‎By these definitions‎, ‎it's shown that‎ ‎finite abelian groups are Frattini supplemented and every conjugate of a‎ ‎Frattini supplement of a subgroup is also a Frattini supplement‎. ‎A group action‎ ‎of a group is defined over the set of Frattini supplements of a normal‎ ‎subgro...

Let H be a subgroup of a finite group G. We say that H is SS-semipermutable in Gif H has a supplement K in G such that H permutes with every Sylow subgroup X of Kwith (jXj; jHj) = 1. In this paper, the Structure of SS-semipermutable subgroups, and finitegroups in which SS-semipermutability is a transitive relation are described. It is shown thata finite solvable group G is a PST-group if and on...