What is Subgroup?

Example 7.8

$\mathbb{U}_n \leq (\C^*, \times)$: nonempty, and for $z, w \in \mathbb{U}_n$, $(zw^{-1})^n = z^n (w^n)^{-1} = 1$. The subgroups of $(\Z, +)$ are exactly the $n\Z$ (proved in Theorem 6.4). An intersection of subgroups is always a subgroup, but a union almost never is (Exercise 7.6).

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.