Group action

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In mathematics, groups often arise as (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.

Group Actions on More General Objects

It is frequently useful to talk about the action of a group on an object besides a set (such as the action of a group on a vector space, a group, a ring, a field, a graph, etc.). This can be done rigorously in the language of category theory. Given a category $\mathcal{C}$ and an object $X$ of $\mathcal{C}$ is a function $\varphi$ from $G$ to $\text{Hom}(X,X)$ such that:

  • For any $g,h\in G$ $\varphi(gh) = \varphi(g)\circ \varphi(h)$
  • $\varphi(1) = 1_X$, where $1$ is the identity element of $G$ and $1_X$ is the identity of $X$.

Notice that, as before, each morphism $\varphi(g)$ has an inverse $\varphi(g^{-1})$, and so $\varphi(g)$ is an automorphism of $X$.

For instance, by this definition, a group action on a vector space, $V$ (also known as a group representation) is a group action on the set of elements of $V$, where each element of the group is associated to a linear function.

Although the above definition is the easiest to understand, it is not necessarily the most useful. To give an alternate definition, we first re-define the notion of a group.

We define a group to be a category with only one object, (say $A$), and in which every morphism is invertible (and is therefore an automorphism of $A$). In terms of our old definition, the 'elements' of the group are actually the morphisms in this category. It is fairly simple to show that definition is equivalent to our old definition.

We can now define an action of a group $G$ to be a functor from $G$ to another category, and we define the action of $G$ on some object $X$ of a category $\mathcal{C}$ to be a functor $\varphi:G\to \mathcal{C}$ such that $\varphi(A) = X$. It follows from the definition of a functor, that this definition is equivalent to the one given above. This definition allows us to easily study the concept of a group action in the framework of category theory. This article is a stub. Help us out by expanding it.

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