Group action

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In mathematics, groups often arise as (a subsets of) the set of permutations of some mathematical object (where the group multiplication law is just composition). Some examples of this would be the symmetric group $S_n$, the alternating group $A_n$, or the dihedral group $D_n$.

We can formalize this notion with the concept of a group action. Loosely speaking, a group action of a group $G$ on a set $X$ is an assignment of a bijection $X\to X$ to each element $g\in G$.

More formally, we can define a group action of a group $G$ on a set $X$ as a function $G\times X\to X$, (where we denote the image of $(g,x)$ by $g\cdot x$) which satisfies the following properties:

  • $g\cdot(h\cdot x) = (gh)\cdot x$ for all $g,h\in G$ and all $x\in X$
  • $1\cdot x = x$ for all $x\in X$ (where $1$ is the identity element of $G$).

Typically, we call $X$ a $G$-set, and we say that $G$ acts on $X$.

Notice that any $g\in G$ indeed gets associated with a function $X\to X$, specifically the function $x\mapsto g\cdot x$. The first property means that the multiplication of two group elements $g$ and $h$ is indeed just the composition of their corresponding functions, and the second property states that the function associated to the identity element it's just the identity function. Notice also that the function associated to $g\in G$ always has an inverse, specifically the function associated to $g^{-1}$. It follows that each group element is in fact associated to a bijection $X\to X$.

Notice that this assignment is not necessarily one-to-one, that is it is possible that two distinct elements of $G$ will be associated to the same function (for instance, it is possible that we would have $g\cdot x = x$ for all $g\in G$ and $x\in X$, and so every element of $G$ would be associated to the identity function). If this assignment is one-to-one (i.e. if for every $g,h\in G$, there is some $x\in X$ for which $g\cdot x \neq h\cdot x$) then we say $G$ acts faithfully on $X$.

Now since the symmetric group $S_X$ is the group of all bijections $X\to X$ we can think of a group action as a homomorphism from $G$ to $S_X$. This homomorphism is injective iff the action is faithful.

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