A question of fully invariant subgroup

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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 ?
 

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  • #2
tiny-tim
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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! :smile:

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? :wink:
 
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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
morphism
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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.
 

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