Group action

You may have been introduced to group theory as 'the study of symmetry'. Common motivating examples are 3D rotations of a sphere, the discrete rotations and/or reflections of an 𝑛-gon, and translations. Group theory studies the algebraic properties of such symmetries.

But note that in our motivating examples, we always started with 'things' on which our 'symmetries' acted: rotations act on spheres, reflections act on n-gons, etc. Group actions formalize this notion.

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Concretely, group actions facilitate a form of scalar multiplication of group elements upon some other set of elements.

Definition

Let $G$ be a group, and $X$ an arbitrary set. A (left) group action is then a function $\alpha :G\times X\to X$ satisfying the following axioms:[cite:@enwiki:1288386141]

Identity

$\alpha (1,x)=x$, where $1$ is $G$'s identity element.

Composition

$\alpha (g,\alpha (h,x))=\alpha (gh,x)$.

Note how the definition only requires multiplication and identity. Similar notions of "action" are easily defined for semigroups or monoids.

These laws will look very familiar to the categorically-inclined. Try rewriting them in a point-free style. }:3

Similarly, a right group action is defined to be a function $\alpha :X\times G\to X$ for which the following laws hold:

Identity

$\alpha (x,1)=x$.

Composition

$\alpha (\alpha (x,g),h)=\alpha (x,gh)$.

A right group action is simply a left group action in the opposite group. It thus typically suffices to study left group actions alone and argue by duality as necessary.

Notation

When the particular action is clear from the context, $\alpha (g,x)$ may be written $g\cdot x$ or $gx$. This notation is inspired by the analogy of group actions defining scalar multiplication against a set.