Endomorphism image implies powering-invariant

From Groupprops
Jump to: navigation, search
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., endomorphism image) must also satisfy the second subgroup property (i.e., powering-invariant subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about endomorphism image|Get more facts about powering-invariant subgroup

Statement

Suppose G is a group and H is a subgroup of G that is an endomorphism image of G, i.e., there exists an endomorphism \sigma of G such that \sigma(G) = H. Then, H is a powering-invariant subgroup of G, i.e., if G is powered over a prime p (i.e., every element of G has a unique p^{th} root, so is H.

Facts used

  1. Endomorphism image implies divisibility-closed
  2. Divisibility-closed implies powering-invariant

Proof

The proof follows directly from Facts (1) and (2).