What is Group morphism?

Example 7.12

$\exp \colon (\R, +) \to (\R_+^*, \times)$ is a morphism ($\eu^{x+y} = \eu^x \eu^y$), bijective (Proposition 4.1): the additive and multiplicative structures are isomorphic — the historical raison d’être of logarithms. Another morphism: $\theta \mapsto \eu^{\iu\theta}$ from $(\R, +)$ onto the unit circle $(\mathbb{U}, \times)$, with kernel $2\pi\Z$.

Example 7.13 (The sign morphism)

The map $s \colon (\R^*, \times) \to (\{\pm1\}, \times)$ sending $x$ to its sign is a morphism: the sign of a product is the product of the signs. Its kernel is $\intoo0{+\infty}$ (a subgroup, as Definition 7.10 promises), its image all of $\{\pm1\}$: surjective, massively non-injective. Two general lessons in miniature. First, a morphism may crush information: $s$ remembers nothing of $x$ but one bit, and that is its virtue — sign arguments are exactly the computations that factor through $s$. Second, morphisms to $\{\pm1\}$ are the simplest “invariants”: the signature of permutations, built in this chapter’s weekend problem, is the same phenomenon on the group $\mathfrak S_n$, and the parity arguments it powers all descend through such a two-valued morphism.