Finite groups with a Sylow 2-subgroup of type M24, II

A REDUCTION THEOREM FOR UNil OF FINITE GROUPS WITH NORMAL ABELIAN SYLOW 2-SUBGROUP

Let F be a finite group with a Sylow 2-subgroup S that is normal and abelian. Using hyperelementary induction and cartesian squares, we prove that Cappell’s unitary nilpotent groups UNil∗(Z[F ];Z[F ],Z[F ]) have an induced isomorphism to the quotient of UNil∗(Z[S];Z[S],Z[S]) by the action of the group F/S. In particular, any finite group F of odd order has the same UNil-groups as the trivial gr...

Finite Groups with p-Sylow Coverings

Reduction of UNil for finite groups with normal abelian Sylow 2-subgroup

Let F be a finite group with a Sylow 2-subgroup S that is normal and abelian. Using hyperelementary induction and cartesian squares, we prove that Cappell’s unitary nilpotent groups UNil∗(Z[F ];Z[F ],Z[F ]) have an induced isomorphism to the quotient of UNil∗(Z[S];Z[S],Z[S]) by the action of the group F/S. In particular, any finite group F of odd order has the same UNil-groups as the trivial gr...

FINITE GROUPS WITH A SYSTEM OF NILPOTENT SUBGROUPS CONTAINING THE SYLOW SUBGROUP By

Let G be a finite group and p an odd prime. By M < G we denote that M is a proper subgroup of G. Put the set Ψ p (G) = {M:M < G, |G : M| 6= a prime power and |G : M|p = 1}. In this paper we investigate the structure of G if every member of Ψ p (G) is nilpotent.

Relative non-Normal Graphs of a Subgroup of Finite Groups

Let G be a finite group and H,K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H , denoted by NH,K, which is a bipartite graph with vertex sets HHK and KNK(H) and two vertices x ∈ H HK and y ∈ K NK(H) are adjacent if xy / ∈ H, where HK =Tk∈K Hk and NK(H) = {k ∈ K : Hk = H}. We determined some numerical invariants and state that when this graph is planar or...