# Abelian implies every subgroup is potentially characteristic

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a abelian group. That is, it states that in a Abelian group (?), every subgroup satisfying the first subgroup property (i.e., Subgroup (?)) must also satisfy the second subgroup property (i.e., Potentially characteristic subgroup (?)). In other words, every subgroup of abelian group is a potentially characteristic subgroup of abelian group.

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This fact is related to: NPC conjecture

View other facts related to NPC conjectureView terms related to NPC conjecture |

## Contents

## Statement

In an Abelian group, every subgroup is a potentially characteristic subgroup: it can be realized as a characteristic subgroup inside some bigger group.

## Related facts

- Finite implies every normal subgroup is potentially characteristic
- Nilpotent implies every normal subgroup is potentially characteristic
- Hypercentral implies every normal subgroup is potentially characteristic

## Facts used

## Proof

The proof follows directly by piecing together facts (1) and (2).