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A group $G$ is said to act on a set $X$ when there is a map $\phi : G\times X\rightarrow X$ such that the following conditions hold for all elements

  1. $\phi(e,x)=x$ where $e$ is the identity element of $G$.

  2. $\phi (g,\phi(h,x))=\phi(gh,x)$ for all $g,h\in G$.

This something I found in a tutorial for group theory. Now my question is what are the elements in X ? Is it all the permutations of all elements or is it the elements itself? And what is $S_{n}$ (not given above, it's so commonly used), which contains $n!$ elements ? I'm terribly confused between $X$ and $S_{n}$ .

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4 Answers 4

up vote 4 down vote accepted

$X$ is just a set; its elements can be anything: they can be numbers, points of the plane, elements of a group/ring/field, other sets, or pretty much anything that you consider valid elements of a set. For the purposes of the definition of group action, $X$ is not assumed to have any structure. So its elements could be $1$, $2$, $3$, and $4$, (so $X=\{1,2,3,4\}$), or they could be house, school, park, and dog (so $X=\{$`house`, `school`, `park`, `dog` $\}$ ), or anything.

The group $S_n$ is the group of all permutations of the set $\{1,2,\ldots,n\}$, which is a group under composition. In fact, you can view $S_n$ and $\{1,2,\ldots,n\}$ as an example of an action: $X$ would be the set $\{1,2,\ldots,n\}$, $S_n$ would be the group $G$, and $\phi\colon G\times X\to X$ would be the function $$\phi\Bigl(\sigma,k\Bigr) = \sigma(k).$$ (Verify that this satisfies the definition of action that you give).

As you will later see, every action of $G$ on $X$ corresponds to a group homomorphism between $G$ and the group $S_X$ of all permutations of the set $X$ (which is not the samething as $X$ itself, just like $S_n$ is not the same as $\{1,2,\ldots,n\}$). And if $X$ and $Y$ have the same number of elements, then $S_X$ is isomorphic to $S_Y$. So if $G$ acts on a set with $n$ elements, then this action will correspond to a group homomorphism $G\to S_X$, and since $S_X$ is isomorphic to $S_n$, we get a homomorphism from $G$ to $S_n$. So the groups $S_n$ play a big role in the theory of group actions.

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Here's an example that might make it clearer. Let's say that $X$ is the set of points in some square, and let's say that $G$ is the group $Z_4 = \{0, 1, 2, 3\}$. The group action of $G$ acting on $X$ will be that if $x$ is some point of the square, then $\phi(n, x)$ is the point on the square that you get after rotating $x$ counterclockwise around the center of the square by $n\cdot90°$.

So for example the group element $1$ acts on points of the square by rotating them a quarter-turn clockwise around the center of the square, and the group element $3$ acts on points by rotating them a quarter-turn counterclockwise.

The identity element for $G$ is 0, and indeed, we have $\phi(0, x) = x$ for each $x$ in the square, since the result of rotating $x$ around the center by 0° is $x$ again.

Similarly, we want $\phi(a, \phi(b, x)) = \phi(a+b, x)$ for each $x$ in the square and each $a,b$ in $Z_4$. This just says that if we rotate first by $b\cdot90°$ and then by $a\cdot90°$ that's the same as doing a single rotation by $(a+b)\cdot90°$, which is correct. So this is indeed an action of $G=Z_4$ on the square $X$.

This is just an example—there will be many other examples that are quite different—but it's a very typical example.


One thing you should notice is that the choice of a square and $Z_4$ match up closely here.

For example, suppose we took $Z_3$ instead of $Z_4$, but we say that $n$ still represents a rotation of $n\cdot90°$. In $Z_3$ we have $1+2 = 0$, so we should have that $\phi(1, \phi(2, x)) = \phi(1+2, x) = \phi(0, x)$ for any $x$ in the square. But this is wrong. Say $x$ is the upper right corner of the square. Then $\phi(2, x)$ is the lower left corner, and $\phi(1, \phi(2,x))$ is the lower right corner. But $\phi(0, x)$ is the upper right corner, not the lower right corner, so this is not an example of a group action.

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Awesome example –  Malice Jul 9 '12 at 20:03
    
can we explain the stabilizer and the Fix() function with the same example ? –  Malice Jul 9 '12 at 20:16
    
The stabilizer subgroup of a point $x$ of the square is the subgroup of $Z_4$ that leaves $x$ unchanged; it's the set of group elements $g$ for which $\phi(x, g) = x$. For this example, the stabilizer subgroup of every point is the trivial group $\{0\}$, except that the stabilizer of the center point of the square is the whole group $Z_4$, since the action of every element leaves the center of the square fixed. I don't know what the ${\mathrm Fix}()$ function is. Is ${\mathrm Fix}(g)$ the set of points left fixed by some element $g$ of the group? –  MJD Jul 9 '12 at 20:21
    
A quick guess . So the stabilizer group will always contain the identity element in it ? –  Malice Jul 10 '12 at 10:59
    
I'm quoting from a paper , Fix(g) denotes the number of elements of X that are fixed by g € G –  Malice Jul 10 '12 at 11:00

$S_X$ for an arbitrary set is the group of permutations of a set $X$, while $S_n$ is the abstract group of permutations of an $n$-element set.

In general context, $X$ is just an arbitrary set. A group action can also be seen as a homomorphism $\varphi:G\to S_X$ from a group into the group of permutations of a given set (and $\varphi(g)\cdot x:=\phi(g,x)$ with your definition).

In practical context, it is often the case that the group action has some addtitional properties, for example, when $X$ is a topological space, we may say that $G$ acts by homeomorphisms on $X$ if for each $g\in G$, $\varphi(g)$ is a homeomorphism. In this case, $\varphi$ can be seen as a homomorphism from $G$ to the group of homeomorphisms of $X$ (which is a subgroup of the group of permutations).

Similarly, if $X$ is a group, $G$ might act by (group) automorphisms, in which case $\varphi$ is into the automorphism group of $X$, if it is a differentiable manifold, it might act by diffeomorphisms, etc.

The most natural examples of group actions are probably the action of $S_X$ on $X$, in which case $\varphi$ is just the identity, and the action of a group on itself by left- or right- translations or by conjugation, where we have $\varphi(x)\cdot y=xy$, $\varphi(x)\cdot y=yx^{-1}$ and $\varphi(x)\cdot y=xyx^{-1}$, respectively.

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It might help to look at examples of group actions that come up in geometry.

For an example of an infinite group, let $X = \mathbb{R}^2$ and let $G = \mathbb{Z}^2$ act on $X$ by vector addition: $(m,n) \cdot (x,y) = (m+x,n+y)$.

For an example of a finite group, let the cyclic group of order 2 act on the 2-sphere so that the nontrivial group element takes each point to its antipodal point.

Once you get the hang of it, you'll realize there are many, many, many examples with many different properties.

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