What is Group?

Example 7.4

$(\Z, +)$, $(\Q, +)$, $(\R, +)$, $(\C, +)$; $(\Q^*, \times)$, $(\R^*, \times)$, $(\C^*, \times)$, $(\mathbb{U}_n, \times)$ (roots of unity, Definition 3.17); the set $\mathfrak{S}(E)$ of bijections of a set $E$ onto itself, under composition — the symmetric group of $E$, non-abelian as soon as $\abs E \geq 3$. Not groups: $(\N, +)$ (no inverses), $(\Z, \times)$ (only $\pm 1$ invertible).

Example 7.6 (The symmetries of a rectangle)

A (non-square) rectangle admits exactly four isometries onto itself: the identity $e$, the horizontal-axis reflection $h$, the vertical-axis reflection $v$, and the half-turn $r$ about the center. Composition makes this four-element set a group: each element is its own inverse ($h^2 = v^2 = r^2 = e$), and the product of any two distinct non-identity elements is the third ($hv = vh = r$: reflecting in both axes is the half-turn). The full table is symmetric, so the group is abelian — yet it is not the same group as the rotations $\mathbb U_4$ of Example 7.15: there, $\iu$ has order $4$, while here every element has order $\leq 2$. Two groups of the same size can thus have genuinely different multiplication structures — the figure below displays both tables side by side. This four-element group returns as $\{\pm1\} \times \{\pm1\}$, and Exercise 7.7 explains why any group with all squares trivial must, like this one, be abelian.

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.