9  Groups

Groups are the simplest structure on this site worth calling a structure at all — one set, one operation, four axioms — and they’re worth taking seriously precisely because of that simplicity: nearly every axiom-based argument you’ll meet later is a more elaborate version of the argument this chapter makes in full.

9.1 The problem, before the definition

Four situations, deliberately unlike each other:

  • The integers, under addition.
  • The nonzero rational numbers, under multiplication.
  • The four rotations of a square (by 0°, 90°, 180°, 270°) that map it onto itself, under “do one, then the other.”
  • The permutations of \(\{1, 2, 3\}\) — the six ways to reorder three labeled objects — under composition.

Each of these lets you combine two things to get a third thing of the same kind, and undo any combination. Rotate a square 90° twice, and you can get back to where you started by rotating 180° more. Add 5, and you can undo it by adding \(-5\). That’s the pattern worth naming: a set with an operation that combines and — crucially — can always be undone.

9.2 The definition

A group is a set \(G\) with a binary operation \(\ast\) satisfying four axioms:

  1. Closure — for all \(a, b \in G\), \(a \ast b \in G\).
  2. Associativity — \((a \ast b) \ast c = a \ast (b \ast c)\) for all \(a, b, c \in G\).
  3. Identity — there exists \(e \in G\) such that \(e \ast a = a \ast e = a\) for all \(a \in G\).
  4. Inverses — for every \(a \in G\), there exists \(a^{-1} \in G\) such that \(a \ast a^{-1} = a^{-1} \ast a = e\).

That’s the entire definition. Nothing here requires \(a \ast b = b \ast a\) — when it does hold for every pair, the group is called abelian (after Niels Abel), and it’s worth noticing that’s a bonus property, not one of the four axioms; the rotations-of-a-square example above is abelian, but the permutations-of-three-objects example is not (rotate-then-reflect a triangle differs from reflect-then-rotate).

9.3 Why these four, and not fewer

Each axiom is load-bearing, and it’s worth seeing what specifically fails without it:

  • Without closure, “\(G\) with operation \(\ast\)” isn’t describing a self-contained system at all — you could combine two elements and land outside the set you started with, and nothing downstream would even be well-defined.
  • Without associativity, \(a \ast b \ast c\) is ambiguous — you’d need to track parentheses forever, and the notion of “just apply the operations in order” would stop making sense. (Notice associativity says nothing about order — that’s commutativity, a separate and optional property.)
  • Without an identity, there’s no reference point meaning “do nothing” — no way to even state what it means to “undo” an operation, since undoing means returning to something, and that something needs a name.
  • Without inverses, closure/associativity/identity together describe a weaker structure called a monoid (the natural numbers under addition are a monoid, not a group — \(3\) has no additive inverse in \(\mathbb{N}\)). Inverses are exactly what upgrades “combine” into “combine and undo.”

9.4 What follows from the definition

A short, genuinely useful list, none of it requiring more than the four axioms: the identity element is unique (if \(e\) and \(e'\) both worked, \(e = e \ast e' = e'\)); each element’s inverse is unique by the same style of argument; and cancellation holds — \(a \ast b = a \ast c\) forces \(b = c\) (combine both sides with \(a^{-1}\) on the left). None of this needed to know anything about what \(G\)’s elements are. That’s not an accident — it’s the entire reason the abstraction is useful: any fact derivable from the four axioms alone is automatically true of every group, from integers to rotations to permutations to structures nobody has thought of yet.

9.5 What a group is NOT

Not necessarily commutative (see above). Not necessarily made of numbers — most groups that matter in later mathematics (symmetry groups, permutation groups, matrix groups) aren’t. And a group is not the same thing as “the set it’s built on” — \(\mathbb{Z}\) is a group under addition but not a group under multiplication (no element but \(\pm 1\) has a multiplicative inverse in \(\mathbb{Z}\)), which is a useful reminder that “group” names a set plus a chosen operation, never a set alone.

9.6 Symmetry and invariance

Groups are, historically, inseparable from the idea of symmetry: the rotations of a square example above generalizes to the full symmetry group of any object — every transformation that leaves it looking unchanged, under composition — and this connection is not incidental decoration, it’s most of why group theory matters in physics and geometry. A conserved quantity in physics (momentum, energy) corresponds, via Noether’s theorem, to a symmetry group of the underlying equations — one of the deepest instances of the invariance thread from the map: what stays the same under a transformation is exactly what a group is built to describe.

The historical route to the formal definition above ran through exactly this kind of example, not the other way around. Évariste Galois used what we would now call permutation groups in the 1830s to settle a centuries-old question — which polynomial equations are solvable by a formula using roots, like the quadratic formula — by studying the symmetries of an equation’s roots, decades before “group” had the abstract four-axiom definition above. That definition was only isolated and generalized later, by Arthur Cayley (1854) and, in essentially its modern abstract form, Heinrich Weber (1893). The concrete examples came first, by decades; the abstraction came after, distilled from what the examples had in common — a clean instance of the historical graph and the formal graph running in different order from the map.

9.7 Where this leads

Groups have one operation. Rings and Fields, next, are built from two interacting operations — exactly what \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), and \(\mathbb{C}\) all have, and what Vector Spaces needs its scalars to supply.