In this paper, we give some new characterizations of finite p-nilpotent groups by using the notion of HC-subgroups and extend several recent results. Mathematics Subject Classification (2010). 20D10, 20D20.

Suppose that G is a finite group and k field of characteristic $$p>0$$ . We consider the complete cohomology ring $${\mathcal {E}}_M^* = \sum _{n \in {\mathbb Z}} \widehat{{\text {Ext}}}^n_{kG}(M,M)$$ show has two distinguished ideals $$I^* \subseteq J^* {\mathcal {E}}_M^*$$ such $$I^*$$ bounded above in degrees, {E}}_M^*/J^*$$ below degree $$J^*/I^*$$ eventually periodic with terms dimension. ...

the proof of the local nilpotence theorem for 4-engel groups was completed by g. havas and m. vaughan-lee in 2005. the complete proof on the other hand is spread over several articles and the aim of this paper is to give a complete coherent linear version. in the process we are also able to make few simplifications and in particular we are able to merge two of the key steps into one.

Recently Havas and Vaughan-Lee proved that 4-Engel groups are locally nilpotent. Their proof relies on the fact that a certain 4-Engel group T is nilpotent and this they prove using a computer and the Knuth-Bendix algorithm. In this paper we give a short hand-proof of the nilpotency of T .

It is shown that all nontrivial elements in higher K-groups of toric varieties over a class of regular rings are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector bundles on affine toric varieties. 1. Statement of the main result The nilpotence conjecture in K-theory of toric varieties, treated in our previous works, asser...