Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

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 . 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 and .

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 .
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 . 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.