Groups & Group Actions

Closure, identity, inverses — and how a group permutes a set.

The idea

A group records what the integers under addition, the symmetries of a square, and the invertible $n \times n$ matrices have in common: an operation that combines two elements, an element that does nothing, and an inverse that undoes each element. Formally, a group is a set $G$ with an operation $\cdot$ satisfying four axioms: closure — the product of two elements of $G$ lies in $G$; associativity — $(ab)c = a(bc)$; an identity $e$ with $ea = ae = a$ for every $a$; and for each $a$ an inverse $a^{-1}$ with $a a^{-1} = a^{-1} a = e$. The axioms do not require $ab = ba$; a group in which that always holds is abelian.

An action of $G$ on a set $X$ assigns to each $g \in G$ a rearrangement of $X$, written $x \mapsto g \cdot x$, compatible with the group structure: $e \cdot x = x$, and $(gh) \cdot x = g \cdot (h \cdot x)$. Each $g$ then permutes $X$, and $X$ splits into orbits: the orbit of a point $x$ is the set of points $g \cdot x$ that the group can move $x$ to. The elements of $G$ that fix $x$, those with $g \cdot x = x$, form a subgroup called the stabilizer of $x$.

Ways to work on it

• Walkthrough. The four group axioms, then orbits and stabilizers via a small group of rotations.
• Practice. Find additive inverses in (ℤ/nℤ, +).
• Hardest. Read a stabilizer and an orbit off a solid's rotation group.

Not sure where to start? Take the ten-question placement test.