‎in this paper we give a new condition for a sylow $p$-subgroup of a finite group to control transfer‎. ‎then it is deduced a characteri-zation of supersoluble groups that can be seen as a generalization of the well known result concerning the supersolubility of finite groups with cyclic sylow subgroups‎. moreover a condition for a normal embedding of a strongly closed $p$-subgroup is given‎. ...

We consider a class of spectral multipliers on stratified Lie groups which generalise the Hormander and include with an oscillatory factor. Oscillating have been examined extensively in Euclidean setting where sharp, endpoint Lp estimates are well known. In group setting, corresponding bounds for oscillating established by several authors but only open range exponents. this paper we establish L...