# Restriction of automorphism to subgroup not implies automorphism

## 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 $H$.

## 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.