Characteristic equals fully invariant in odd-order abelian group

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a odd-order Abelian group. That is, it states that in a Odd-order Abelian group (?), every subgroup satisfying the first subgroup property (i.e., Characteristic subgroup (?)) must also satisfy the second subgroup property (i.e., Fully characteristic subgroup (?)). In other words, every characteristic subgroup of odd-order Abelian group is a fully characteristic subgroup of odd-order Abelian group.
View all subgroup property implications in odd-order Abelian groups

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View all subgroup property non-implications in odd-order Abelian groups

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View all subgroup property implications

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View all subgroup property non-implications

Statement

In an odd-order Abelian group, a subgroup is characteristic if and only if it is fully characteristic.

Related facts

• Fully characteristic implies characteristic
• Characteristic not implies fully characteristic
• Characteristic not implies fully characteristic in finitely generated Abelian group
• Characteristic not implies fully characteristic in finite Abelian group
• Classification of fully characteristic subgroups in finitely generated Abelian groups

Proof

Given: An odd-order Abelian group $G$.

To prove: A subgroup $H$ of $G$ is characteristic in $G$ if and only if $H$ is fully characteristic in $G$.

Proof: We give the proof in two steps.