Restriction of automorphism to subgroup not implies automorphism

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Statement

We can have a group G, a subgroup H, and an automorphism \sigma of G such that \sigma(H) \subseteq H, but \sigma(H) is not equal to H. In other words, \sigma restricts to an endomorphism of H (which is necessarily an injective endomorphism), but the restriction is not an automorphism of H.

Related facts

Proof

Example of the integers and the rationals

Let G be the group (\mathbb{Q},+), i.e., the additive group of rational numbers. Let H be the subgroup (\mathbb{Z},+), i.e., the additive group of integers. Consider the automorphism \sigma of \mathbb{Q} given by x \mapsto 2x, i.e., it sends every rational number to its double.

\sigma is clearly an automorphism of \mathbb{Q}, and \sigma sends \mathbb{Z} to within itself: the restriction of \sigma to \mathbb{Z} is the map that sends every integer to its double. The restriction of \sigma to \mathbb{Z} is not an automorphism of \mathbb{Z}: the image \sigma(\mathbb{Z}) is the proper subgroup comprising even integers.