let $g={rm sl}_2(p^f)$ be a special linear group and $p$ be a sylow‎ ‎$2$-subgroup of $g$‎, ‎where $p$ is a prime and $f$ is a positive‎ ‎integer such that $p^f>3$‎. ‎by $n_g(p)$ we denote the normalizer of‎ ‎$p$ in $g$‎. ‎in this paper‎, ‎we show that $n_g(p)$ is nilpotent (or‎ ‎$2$-nilpotent‎, ‎or supersolvable) if and only if $p^{2f}equiv‎ ‎1,({rm mod},16)$‎.

We study the residual properties of geometric 3–manifold groups. In particular, we study conditions under which geometric 3–manifold groups are virtually residually p for a prime p, and conditions under which they are residually torsion–free nilpotent. We show that for every prime p, every geometric 3–manifold group is virtually residually p. We show that geometric 3– manifold groups are virtua...

Suppose K is a finite field extension of Qp containing a primitive p-th root of unity. Let ΓK(1) be the Galois group of a maximal p-extension of K with the Galois group of period p and nilpotent class < p. In the paper we describe the ramification filtration {ΓK(1)}v>0 and relate it to an explicit form of the Demushkin relation for ΓK(1). The results are given in terms of Lie algebras attached ...

We show that ω-categorical rings with NIP are nilpotent-by-finite. We prove that an ω-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an ω-categorical group with at least one strongly regular type is abelian. Moreover, we get that each ω-categorical, characteristically simple p-group with NIP has an infinite, definable abelian subgroup. Assuming additionally the e...

A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalization of a theorem of Baer for the small class case. The approach is also used to obtain some recent results on the capability of certain nilpotent groups of class 2. We also prove a necessary condition for the capability of an arbitrary p-group of ...

Let KG be the group ring of a group G over a field of characteristic p > 0, p ^ 2, 3. Suppose G contains no element of order p (if p > 0). Group algebras KG with unit group U(KG) solvable and n-Engel are characterized. Let ATG be the group ring of a group G over a field K of characteristic p > 0 and let U(KG) denote its group of units. Several authors including Bateman [1], Bateman and Coleman ...

The genus of a finitely generated nilpotent group G is defined as the set of isomorphism classes of finitely generated nilpotent groups K such that the p-localizations Kp, Gp are isomorphic for all primes p [19]. This notion turns out to be particularly relevant in the study of non-cancellation phenomena in group theory and homotopy theory. In the above definition, the restriction of finite gen...

A finite non-abelian group G is called metahamiltonian if every subgroup of either abelian or normal in G. If non-nilpotent, then the structure has been determined. nilpotent, determined by its Sylow subgroups. However, classification p-groups an unsolved problem. In this paper, are completely classified up to isomorphism.

Conditions are given for a class 2 nilpotent group to have no central extensions of class 3. This is related to Betti numbers and to the problem of representing a class 2 nilpotent group as the fundamental group of a smooth projective variety. Surveys of the work on the characterization of the fundamental groups of smooth projective varieties and Kähler manifolds (see [1],[3], [9]) indicate tha...