Automorphism

Formal Definition

An isomorphism of a group with itself.

Informal Definition

The operation of $G \mapsto G$, where $G$ is a group, is isomorphic.

Example(s)

The map $\mu: {Z} \rightarrow {Z}$ given by $\mu(x) = -x$ . For ${Z}$ is cyclic, and an isomorphism ${Z}\rightarrow {Z}$ must carry a generator to a generator. Since the only generators of ${Z}$ are 1 and -1, the only automorphisms are the maps sending $1 \mapsto 1$ and $1 \mapsto -1$.

Non-example(s)

${Q} \mapsto {Q}$ and ${R} \mapsto {R}$.