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Formal Definition

An isomorphism of a group with itself.

Informal Definition

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


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 $1 \mapsto -1$.


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

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