suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...

A subgroup H of a group G is contranormal if HG=G. In finite groups, there are no proper subgroups, then the nilpotent but this not true in infinite groups as well-known Heineken–Mohamed show. We call such without subgroups “contranormal-free.” article, we prove various results concerning contranormal-free proving, for example that locally generalized radical which have section rank hypercentral.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...

‎In the classical group theory there is‎ an open question‎: ‎Is every torsion free n-Engel group (for n ≥ 4)‎, nilpotent?‎. ‎To answer the question‎, ‎Traustason‎ [11] showed that with some additional conditions all‎ ‎4-Engel groups are locally nilpotent‎. ‎Here‎, ‎we gave some partial‎ answer to this question on Engel fuzzy subgroups‎. ‎We show that if μ is a normal 4-Engel fuzzy‎ subgroup of ...

Lattices and parabolic subgroups are the obvious examples of cocompact subgroups of a connected, semisimple Lie group with finite center. We use an argument of C. C. Moore to show that every cocompact subgroup is, roughly speaking, a combination of these. We study a cocompact subgroup H of a connected, semisimple Lie group G with finite center. The case where H is discrete is very important and...

The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of...

IN this paper we shall study the connection between local and global nilpotency for groups which satisfy certain chain conditions. In the first section, we shall prove that a group satisfying the chain condition on centralizers has a unique maximal normal nilpotent subgroup (the Fitting subgroup), i.e. the group generated by all normal nilpotent subgroups must again be nilpotent. This generaliz...

Let G be a permutation group of set ? and k positive integer. The k-closure is the greatest (w.r.t. inclusion) subgroup G(k) in Sym(?) which has same orbits as under componentwise action on ?k. It proved that finite nilpotent coincides with direct product k-closures all its Sylow subgroups.

Consider a finite group G and subgroups H,K of G. We say that H and K permute if HK = KH and call H a permutable subgroup if H permutes with every subgroup of G. A group G is called quasi-Dedekind if all subgroups of G are permutable. We can define, for every finite group G, an arithmetic quantity that measures the probability that two subgroups (chosen uniformly at random with replacement) per...

Let G be a finite almost simple group with socle G0. A (nontrivial) factorization of is an expression the form G=HK, where factors H and K are core-free subgroups. There extensive literature on factorizations groups, important applications in permutation theory algebraic graph theory. In recent paper, Xia second-named author describe groups solvable factor H. Several infinite families arise con...