Endomorphism image implies powering-invariant
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)
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Suppose is a group and is a subgroup of that is an endomorphism image of , i.e., there exists an endomorphism of such that . Then, is a powering-invariant subgroup of , i.e., if is powered over a prime (i.e., every element of has a unique root, so is .
The proof follows directly from Facts (1) and (2).