# Characteristically simple implies CSCFN-realizable

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., characteristically simple group) must also satisfy the second group property (i.e., CSCFN-realizable group)

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## Contents

## Statement

Any characteristically simple group can be realized as a CSCFN-subgroup (?) of some group.

## Related facts

- Characteristically simple and non-abelian implies automorphism group is complete
- Additive group of a field implies monolith in holomorph

## Facts used

- Characteristically simple implies center is a direct factor
- Center is a direct factor implies NSCFN-realizable
- Characteristically simple and NSCFN implies monolith
- Monolith is characteristic

## Proof

The proof follows from facts (1)-(4).