Representation Theory/Set representations

Definition (set representation):

A set representation is a representation of a group in the category of sets.

Alternatively, set representations are also called group actions, and we say that $G$ acts on a set $S$. Whenever $g\in G$, we will denote the corresponding element of $\operatorname {Aut} (S)$ (which are just the permutations of $S$) by $g$ as well, so that $g$ becomes a bijective function on $S$. In particular, for $x\in S$, we can make sense of expressions such as $gx$ (which shall be a shorthand for $g(x)$).

Definition (orbit):

Let $G$ be a group acting on a set $S$, and let $x\in S$. The orbit of the element $x$ is defined to be the set

$Gx:=\{gx|g\in G\}$.

Proposition (group action partitions set into orbits):

Let $G$ be a group acting on a set $S$. Then $S$ is partitioned into orbits of $G$.

Proof: We prove the claim by proving that being in the same orbit is an equivalence relation.

1. $x\sim x$, since $ex=x$ as the representation is a group homomorphism, so that $e$ represents the identity
2. $x\sim y\Rightarrow y\sim x$ since $gx=y\Rightarrow x=g^{-1}y$ as the representation is a group morphism, so that $g^{-1}gx=x$
3. $x\sim y\wedge y\sim z\Rightarrow x\sim z$ since $y=gx$ and $z=hy$ imply $z=(hg)x$ since $(hg)x=h(gx)$ as the representation is a group homomorphism $\Box$

Definition (transitive action):

Suppose that $G$ is a group acting on $S$. This action is called transitive if and only if all elements are in the same orbit, ie. there is only one orbit, and it is all of $S$.

Equivalently, we could have required that for all $x,y\in S$, there exists $g\in G$ such that $gx=y$.