# Cyclic over central implies abelian

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., cyclic group) must also satisfy the second group property (i.e., epabelian group)

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## Contents

## Statement

### Straightforward formulation

Suppose are groups, such that is a central subgroup of (in other words, is contained in the center of ), and is cyclic. Then is an abelian subgroup of , i.e., it is Abelian as a group.

### In terms of cyclic and epabelian groups

Any cyclic group is an epabelian group.

## Related facts

### Applications

- A nontrivial cyclic group is not a capable group: it cannot be realized as the quotient of a group by its center.
`For full proof, refer: Cyclic and capable implies trivial` - Characteristically metacyclic and commutator-realizable implies abelian

## Proof

**Given**: A group , subgroups . is in the center of , and is cyclic.

**To prove**: is abelian.

**Proof**: Suppose is a generator of and is an element of whose image mod is . Then, we have contains and intersects every coset of in . Hence, .

- is in the center of : This follows from the fact that is in the center of .
- is in the center of : The centralizer of in contains , and also contains , since is in the center of . Hence, the centralizer of contains , so is in the center of .
- The center of is , and hence is abelian: From the previous two steps, is in the center of , which in turn is contained in . But , so equals its own center.