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From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
View a complete list of pages giving proofs of equivalence of definitions

Contents 1 Statement 2 Related facts 2.1 Stronger facts 2.2 Other related facts 3 Facts used 4 Proof

Statement

The following are equivalent for a subgroup H of a group G :

1. H is a normal subgroup of G.
2. H is a potentially characteristic subgroup of G in the following sense: there exists a group K containing G such that H is a characteristic subgroup of K.

Related facts

Stronger facts

• NRPC theorem

Other related facts

• Finite NPC theorem
• Finite NIPC theorem
• Fact about amalgam-characteristic subgroups: finite normal implies amalgam-characteristic, periodic normal implies amalgam-characteristic, central implies amalgam-characteristic

Facts used

1. Characteristicity is centralizer-closed

Proof

Given: A group G, a normal subgroup H of G.

To prove: There exists a group K containing G such that H is characteristic in K.

Proof:

1. Let S be a simple non-abelian group that is not isomorphic to any subgroup of G: Note that such a group exists. For instance, we can take the finitary alternating group on any set of cardinality strictly bigger than that of G.
2. Let K be the restricted wreath product of S and G, where G acts via the regular action of G / H and let V be the restricted direct power S^{G / H}. In other words, K is the semidirect product of the restricted direct power V = S^{G / H} and G, acting via the regular group action of G / H.
3. Any homomorphism from V to G is trivial: By definition, V is a restricted direct product of copies of S. Since S is simple and not isomorphic to any subgroup of G, any homomorphism from S to G is trivial. Thus, for any homomorphism from V to G is trivial.
4. V is characteristic in K: Under any automorphism of K, the image of V is a homomorphic image of V in K. Its projection to is a homomorphic image of V in G, which is trivial, so the image of V in K must be in V.
5. The centralizer of V in K equals H: By definition, H centralizes V. Using the fact that S is centerless and that inner automorphisms of S cannot be equal to conjugation by elements in , we can show that it is precisely the center.
6. H is characteristic in K: This follows from the previous two steps and fact (1).