Groupprops, The Group Properties Wiki (pre-alpha)

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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., characteristically simple group) must also satisfy the second group property (i.e., CSCFN-realizable group)
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Contents 1 Statement 2 Related facts 3 Facts used 4 Proof

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

1. Characteristically simple implies center is a direct factor
2. Center is a direct factor implies NSCFN-realizable
3. Characteristically simple and NSCFN implies monolith
4. Monolith is characteristic

Proof

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