Restriction of automorphism to subgroup invariant under it and its inverse is automorphism

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Statement

Suppose G is a group, H is a subgroup, and σ is an automorphism of G such that both σ and σ − 1 leave H invariant. Then, σ restricts to an automorphism of H.

Related facts

  • Restriction of automorphism to subgroup not implies automorphism: If σ is an automorphism of a group G and H is a subgroup of G such that \sigma(H) \subseteq H. The restriction of σ to H need not be an automorphism of H.
  • For any group-closed automorphism property (i.e., any property of automorphisms such that for any given group the automorphisms satisfying the property form a group), any subgroup invariant under all automorphisms with the property satisfies the additional condition that the restriction of each such automorphism to the subgroup is an automorphism of the subgroup. A subgroup property that can be expressed this way is termed an auto-invariance property. Examples of this are:
    • The property of being a characteristic subgroup is the invariance property with respect to all automorphisms. The restriction of any automorphism of the whole group to a characteristic subgroup is an automorphism of the subgroup.
    • The property of being a normal subgroup is the invariance property with respect to all inner automorphisms. The restriction of any inner automorphism of the whole group to a normal subgroup is an automorphism of the subgroup.

Proof

Given: A group G, a subgroup H, an automorphism σ of G such that \sigma(H) \subseteq H and \sigma^{-1}(H) \subseteq H.

To prove: σ(H) = H and the restriction of σ to H is an automorphism of H.

Proof:

  1. Since σ is an automorphism of G, so is σ − 1, and their composite (both ways) is the identity map on G. In other words, σ(σ − 1(g)) = g and σ − 1(σ(g)) = g for all g \in G.
  2. By our assumption, the restrictions σ | H and σ − 1 | H are both functions from H to itself. Further, we have that σ(σ − 1(h)) = h and σ − 1(σ(h)) = h for all h \in H. Thus, σ | H and σ − 1 | H are two-sided inverses of each other, and are thus both bijections. In particular, σ(H) = H.
  3. Finally, since σ is a homomorphism, so is σ | H. Thus, σ | H is a bijective homomorphism from H to itself, and is hence an automorphism of H.
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