1. In [3], we have shown that in a finite ZP-group G in which A and B are cyclic and A is its own normalizer, the commutator subgroup T of G is cyclic and G = AT with AC\T= I. This result can be used to determine the structure of arbitrary ZP-groups in which A and B are cyclic. If A is a subgroup of a group G, define the subgroup Ni(A) of G inductively by the formula Ni(A)=No(Ni~1(A)), and deno...

In this paper, we prove the p-nilpotency of a finite group with assumption that some subgroups of Sylow subgroup are weakly s-semipermutable subgroups in the normalizer of Sylow subgroups. Our results unify and generalize some earlier results. Mathematics Subject Classification: 20D10, 20D15

Let P be an extraspecial p-group which is neither dihedral of order 8, nor of odd order p and exponent p. Let G be a finite group having P as a Sylow p-subgroup. Then the mod-p cohomology ring of G coincides with that of the normalizer NG(P ).

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, the nilpotent case. It turns out that context, like for finite nilpotency is equivalent to normalizer property or uniqueness of Sylow subgroups, provided maximal normal torsion-free subgroup nilpotent. As consequence, we show decompositions prove Lie group an expansion reals if...

In this thesis we primarily consider saturated fusion systems which are of essential rank 1, i.e. which contain only one conjugacy class of essential subgroups. We prove that they all can be realized by an amalgam of two finite groups. This leads us to the application of amalgam related methods like pushing up techniques and results about FF-modules. As a first key result, in fusion systems of ...

In this paper we study the normalizer decomposition of a compact Lie group G using information fusion system F on maximal discrete p-toral subgroup. We prove that there is an injective map from set conjugacy classes chains F-centric, F-radical subgroups to p-centric, p-stubborn continuous subgroups. The bijection when π0(G) finite p-group. also classifying space chain mod p equivalent correspon...

This note shows how to construct a representation of the automorphic extension G of the simple group S = O 8 (5) by a symmetric group on three points, together with an embedding of the normalizer H of an O 8 (2) type subgroup of O + 8 (5). As an application, it is shown that the permutation representation of G on the cosets of H has a base of length two. This question arose in [BGS].

Let 1 → (K, K1) → (G, NG(K1)) → (Q, Q1) → 1 be a short exact sequence of pairs of finitely generated groups with K strongly hyperbolic relative to the subgroup K1. Let relative hyperbolic boundary of K with respect to K1 contains atleast three distinct parabolic end points and for all g ∈ G there exists k ∈ K such that gK1g −1 = kK1k , then we prove that there exists a quasi-isometric section s...

Let G be a connected reductive linear algebraic group. We use geometric methods to investigate G-completely reducible subgroups of G, giving new criteria for G-complete reducibility. We show that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G; this allows us to use ideas of R.W. Richardson and Hilbert– Mumford–Kempf from geometric invariant theory. We ded...