# Abelian implies self-centralizing in holomorph

From Groupprops

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., abelian group) must also satisfy the second group property (i.e., NSCFN-realizable group)

View all group property implications | View all group property non-implications

Get more facts about abelian group|Get more facts about NSCFN-realizable group

## Statement

Suppose is an abelian group and is its holomorph. Then, . In other words, is a Self-centralizing subgroup (?) of .

Further, since every group is normal fully normalized in its holomorph, this tells us that is normal, self-centralizing and fully normalized in its holomorph. In short, is a NSCFN-subgroup (?) of its holomorph, and thus, is a NSCFN-realizable group (?).

## Related facts

### Applications

- Center is a direct factor implies NSCFN-realizable
- Characteristically simple implies NSCFN-realizable
- Centerless implies NSCFN-realizable
- Characteristically simple implies monolith in automorphism group

## Proof

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]