Group actions: What are they?

Question

Im currently studying pure maths and I am unsure as to what a group action is? I understand that a group action is defined as:

For a g $\in$ G, let G be a group and for a s $\in$ S be a set, then a group action is:

$\rhd$ : G $\times$ S $\rightarrow$ S

So for example what does: g $\rhd$ S mean? And then g$_1$ $\rhd$ ( g$_2 \rhd$ S )

Also what is the test for finding group actions?

Answer 1

Let $G$ be a group and $S$ be a set. One says that $G$ acts on $S$ if and only if there exists a map $\cdot:G\times S\rightarrow S$ such that for all $s\in S,1_G\cdot s=s$ ($1_G$ acts trivially on S) and for all $g_1,g_2\in G$, $g_1\cdot(g_2\cdot s)=(g_1g_2)\cdot s$ (the action is compatible with the group structure of $G$). Usually one write $g\cdot s$ instead of $\cdot(g,s)$.

Your definition of group action is laking the last two properties.

An equivalent definition of group action is a group morphism from $G$ to $\mathfrak{S}(S)$, the group of bijections of $S$.

Let us examine some examples.

• A group $G$ acts on itself by left translation through $(g,h)\mapsto gh$.

• A group $G$ acts on itself by right translation through $(g,h)\mapsto hg^{-1}$ (try to understand why I chose to multiply by $g^{-1}$ instead of $g$).

• A group $G$ acts on itself by conjugation through $(g,h)\mapsto ghg^{-1}$.

• $\mathfrak{S}_n$ acts on $\{1,\cdots,n\}$ through $(\sigma,i)\mapsto\sigma(i)$.

• Let $E$ be a $k$-vector space, $\textrm{GL}(E)$ acts on $E$ through $(g,x)\mapsto g(x)$.

Let me try to motivate the definition of group action. If $E$ is a $k$-vector space, then $k$ acts on $E$ through scalar multiplication. In that sense, the notion of group action is a way to generalize the definition of vector space.

Answer 2

You are given a group $G$ and a set $S$. Note that the set of permutations of $S$ is also group (of order n! if $S$ happens to be finite, of cardinality $n$. But $S$ could very well be infinite).

If there is a way of associating to each element of $G$ a permutation of $S$ we say $G$ acts on $S$: such as association must be homomorphic.

Often people write $g.s$ to mean the effect of the permutation associated with $g$ on the element $s\in S$. Homomorphic condition translates to (in this notation) $g_1.(g_2.s)= (g_1g_2).s$ and $e.s=s$ for all $s\in S, g_1,g_2\in S$.

Consider the infinite set of points on the plane. Among all permutations of this infinite set consider the following set of permutations $T_n$ defined for each integer: $T_n{x\choose y}= {x+ny\choose y} $. Note that these permutations shuffle points horizontally, and so of very restricted kind.

Now check that the permutation $$T_{n+m}= T_n\circ T_m\qquad T_0=\rm identity\hskip2in(*) $$ Thus we have associated one permutation $T_n$ for each integer $n$, and it is homomorphic association, by $(*)$.

So we can say the group of integers Z acts on the set of points on the plane.

Exercise: can you give a vertically shuffling action in the same setup?