A question of fully invariant subgroup

[rulin]

A subgroup H of a group G is fully invariant if t(H)<=H for every endomorphism t of G. Let G is finite p-group has a fully invariant subgroup of order d for every d dividing |G|. What is the structure of G ?

 

[tiny-tim]

Welcome to PF!

A subgroup H of a group G is fully invariant if t(H)<=H for every endomorphism t of G. Let G is finite p-group has a fully invariant subgroup of order d for every d dividing |G|. What is the structure of G ?

Hi rulin ! Welcome to PF!

Hint: if p and q are different prime factors of |G|, and their fully invariant subgroups are P and Q, then what is the order of the subgroup generated by P and Q?

 

[rulin]

Maybe i don't know the order of the subgroup generated by P and Q, but this group satisfied above condition must be nilpotent.

 

[morphism]

Well that doesn't say much, because any finite p-group is nilpotent.

To be honest, I don't understand the relevance of tiny-tim's hint.