Equivalence of definitions of transitively normal subgroup

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

Statement

The following are equivalent for a normal subgroup K of a group G:

  1. For any normal subgroup H of K, H is a normal subgroup of G.
  2. For any normal automorphism \sigma of G, the restriction of \sigma to K is also a normal automorphism of K.

Definitions used

Normal automorphism

Further information: Normal automorphism (?)

An automorphism \sigma of a group G is termed a normal automorphism if, for every normal subgroup N of G, \sigma restricts to an automorphism of N.

Proof

(1) implies (2)

Given: A group G, a subgroup K of G such that any normal subgroup H of K is also a normal subgroup of G. \sigma is a normal automorphism of G.

To prove: \sigma restricts to an automorphism \sigma' of K that is a normal automorphism of K.

Proof:

  1. K is normal in G: Since K is a normal subgroup of itself, setting H = K in the condition on K yields that K is normal in G.
  2. \sigma restricts to an automorphism \sigma' of K: This follows from the previous step and the definition of normal automorphism.
  3. For any normal subgroup H of K, H is a normal subgroup of G: This is by assumption.
  4. For any normal subgroup H of K, \sigma' restricts to an automorphism of H: By the previous step, H is normal in G, so, by assumption, \sigma restricts to an automorphism of H. But since H \le K \le G and the restriction of \sigma to K is \sigma', the restriction of \sigma to H equals the restriction of \sigma' to H.

From the last step, \sigma' is a normal automorphism of K, completing the proof.

(2) implies (1)

Given: A group G, a subgroup K of G such that every normal automorphism of G restricts to a normal automorphism of K. H is a normal subgroup of K.

To prove: H is a normal subgroup of G, i.e., for any g \in G and h \in H, ghg^{-1} \in H.

Proof: Let \sigma = c_g = x \mapsto gxg^{-1} be conjugation by g (i.e., an inner automorphism).

  1. \sigma = c_g is a normal automorphism of G: By the definition of normal subgroup, c_g restricts to an automorphism of every normal subgroup of G. Thus, c_g is a normal automorphism of G.
  2. The restriction of c_g to K, which we call \sigma', is a normal automorphism of K: This follows from the given data for K.
  3. \sigma' and hence, \sigma, restricts to an automorphism of H: Since H is a normal subgroup of K, and \sigma' is a normal automorphism of K, \sigma' restricts to an automorphism of H. Hence, \sigma restricts to an automorphism of H.
  4. \sigma(h) \in H, i.e., c_g(h) = ghg^{-1} \in H: This follows immediately from the previous step.