Restriction of automorphism to subgroup invariant under it and its inverse is automorphism
- Restriction of automorphism to subgroup not implies automorphism: If is an automorphism of a group and is a subgroup of such that . The restriction of to need not be an automorphism of .
- 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.
Given: A group , a subgroup , an automorphism of such that and .
To prove: and the restriction of to is an automorphism of .
- Since is an automorphism of , so is , and their composite (both ways) is the identity map on . In other words, and for all .
- By our assumption, the restrictions and are both functions from to itself. Further, we have that and for all . Thus, and are two-sided inverses of each other, and are thus both bijections. In particular, .
- Finally, since is a homomorphism, so is . Thus, is a bijective homomorphism from to itself, and is hence an automorphism of .