Let C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C are torsion-by-nilpotent, then all soluble groups of C are torsion-by-nilpotent. From that, we deduce the following consequence, similar to a well-known result of P. Hall: if H is a normal subgroup of a group G such that H and G/H ′ are (locally finite)-by-nilpotent, then G is (l...

Let G be a finite group and assume that of automorphisms A is acting on such have coprime orders. Recall subgroup H said to TI-subgroup if it has trivial intersection with its distinct conjugates in G. We study the solubility other properties when we certain invariant subgroups are TI-subgroups, precisely all A-invariant subgroups, non-nilpotent non-abelian G, respectively, TI-subgroups.

If G is a group with supersimple theory having finite SU -rank, the subgroup of G generated by all of its normal nilpotent subgroups is definable and nilpotent. This answers a question asked by Elwes, Jaligot, Macpherson and Ryten in [5]. If H is any group with supersimple theory, the subgroup of H generated by all of its normal soluble subgroups is definable and soluble.

In this paper, we prove the conjectures made in a joint paper of the author with Carlson and Robinson, on the vanishing of cohomology of a nite group G. In particular, we prove that if k is a eld of characteristic p, then every non-projective kG-module M in the principal block has nontrivial cohomology in the sense that H (G;M) 6= 0, if and only if the centralizer in G of every element of order...

We show that free products of finitely generated and residually p-finite nilpotent groups, amalgamating p-closed central subgroups are residually p-finite. As a consequence, we are able to show that generalized free products of residually p-finite abelian groups are residually p-finite if the amalgamated subgroup is closed in the pro-p topology on each of the factors.

In this paper, we prove the conjectures made in a joint paper of the author with Carlson and Robinson, on the vanishing of cohomology of a nite group G. In particular, we prove that if k is a eld of characteristic p, then every non-projective kG-module M in the principal block has nontrivial cohomology in the sense that H (G; M) 6 = 0, if and only if the centralizer in G of every element of ord...

This note explains consequences of recent work of Frank Quinn for computations of Nil groups in algebraic K-theory, in particular the Nil groups occurring in the K-theory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group. 1. Statement of Results Let R be a ring with unit. For an integer q, let KqR be the algebraic K-group of Bass and Quillen. Bass ...

Abstract Following J. S. Rose, a subgroup ???? of group ???? is said to be contranormal in if G = H G=H^{G} . It well known that finite nilpotent and only it has no proper subgroups. We study nilpotent-by-?ernikov groups with Furthermore, we the structure ...

we call $h$ an $ss$-embedded subgroup of $g$ if there exists a‎ ‎normal subgroup $t$ of $g$ such that $ht$ is subnormal in $g$ and‎ ‎$hcap tleq h_{sg}$‎, ‎where $h_{sg}$ is the maximal $s$-permutable‎ ‎subgroup of $g$ contained in $h$‎. ‎in this paper‎, ‎we investigate the‎ ‎influence of some $ss$-embedded subgroups on the structure of a‎ ‎finite group $g$‎. ‎some new results were obtained.

Abstract A subgroup functor τ \tau is said Φ \Phi -regular if for all primitive groups G G , whenever H ∈ ( ) H\in \left(G) a p p -subgroup and N N minimal normal of then <m:mo...