A few more facts about group actions

There’s another thing I should have mentioned before. When a group $G$ acts on a set $S$ , there is a bijection between the orbit of a point $x$ and the set of cosets of $G_x$ in $G$ . In fact, $gx=hx$ if and only if $h^{-1}gx=x$ if and only if $h^{-1}g$ is in $G_x$ if and only if $gG_x=hG_x$ . This is the the bijection we need.

This has a few immediate corollaries. Yesterday, I mentioned the normalizer $N_G(K)$ of a subgroup $K$ . When a subgroup $H$ acts on $G$ by conjugation we call the isotropy group of an element $x$ of $G$ the “centralizer” $C_H(x)$ of $x$ in $H$ . This gives us the following special cases of the above theorem:

The number of elements in the conjugacy class of $x$ in $G$ is the number of cosets of $C_G(x)$ in $G$ .
The number of subgroups conjugate to $K$ in $G$ is the number of cosets of $N_G(K)$ in $G$ .

In fact, since we’re starting to use this “the number of cosets” phrase a lot it’s time to introduce a bit more notation. When $H$ is a subgroup of a group $G$ , the number of cosets of $H$ in $G$ is written $\left[G:H\right]$ . Note that this doesn’t have to be a finite number, but when $G$ (and thus $H$ ) is finite, it is equal to the number of elements in $G$ divided by the number in $H$ . Also notice that if $H$ is normal, there are $\left[G:H\right]$ elements in $G/H$ .

This is why we could calculate the number of permutations with a given cycle type the way we did: we picked a representative $g$ of the conjugacy class and calculated $\left[S_n:C_{S_n}(g)\right]$ .

One last application: We call a group action “free” if every element other than the identity has no fixed points. In this case, $G_x$ is always the trivial group, so the number of points in the orbit of $x$ is $\left[G:G_x\right]$ is the number of elements of $G$ . We saw such a free action of Rubik’s Group, which is why every orbit of the group in the set of states of the cube has the same size.

February 24, 2007 - Posted by John Armstrong | Algebra, Group Actions, Group theory

5 Comments »

I get four “formula does not parse” errors. Looking at the ALT text, I’d guess that these are due to that bug where WordPress decided that brackets had to be done as \left[ and \right].

Comment by Blake Stacey | March 6, 2008 | Reply
Thanks. Got ’em.

Comment by John Armstrong | March 6, 2008 | Reply
[…] of our set by rearranging the last elements of the permutation. What’s more, it acts freely — with no fixed points — so every orbit has the same size: . But since we only care […]

Pingback by Permutations and Combinations « The Unapologetic Mathematician | December 29, 2008 | Reply
[…] also know from some general facts about group actions that the number of elements in the conjugacy class is equal to the number of cosets of the […]

Pingback by Conjugates « The Unapologetic Mathematician | September 10, 2010 | Reply
[…] as for some . That is, the set of all Young tabloids is the orbit of the canonical one. By general properties of group actions we know that there is a bijection between the orbit and the index of the stabilizer of in . That […]

Pingback by Dimensions of Young Tabloid Modules « The Unapologetic Mathematician | December 16, 2010 | Reply

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