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.

It is well known that the set of values a lower central word in group G need not be subgroup. For fixed ?r and for p ? 5, Guralnick showed if finite p-group such verbal subgroup ?r(G) abelian 2-generator, then consists only ?r-values. In this paper we extend result, showing assumption can dropped. Moreover, show result remains true even p= 3. Finally, prove analogous pro-p groups true.

A finite p-group S is said to be of supersoluble type if every fusion system over supersoluble. The main aim this paper characterise the p-groups type. Abelian and metacyclic are completely described. Furthermore, we show that Sylow p-subgroups a simple group must cyclic.