# A question of fully invariant subgroup

1. Sep 4, 2008

### 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 ?

2. Sep 5, 2008

### tiny-tim

Welcome to PF!

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?

3. Sep 5, 2008

### 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.

4. Sep 7, 2008

### 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.