A few more facts about group actions
There’s another thing I should have mentioned before. When a group acts on a set
, there is a bijection between the orbit of a point
and the set of cosets of
in
. In fact,
if and only if
if and only if
is in
if and only if
. This is the the bijection we need.
This has a few immediate corollaries. Yesterday, I mentioned the normalizer of a subgroup
. When a subgroup
acts on
by conjugation we call the isotropy group of an element
of
the “centralizer”
of
in
. This gives us the following special cases of the above theorem:
- The number of elements in the conjugacy class of
in
is the number of cosets of
in
.
- The number of subgroups conjugate to
in
is the number of cosets of
in
.
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 is a subgroup of a group
, the number of cosets of
in
is written
. Note that this doesn’t have to be a finite number, but when
(and thus
) is finite, it is equal to the number of elements in
divided by the number in
. Also notice that if
is normal, there are
elements in
.
This is why we could calculate the number of permutations with a given cycle type the way we did: we picked a representative of the conjugacy class and calculated
.
One last application: We call a group action “free” if every element other than the identity has no fixed points. In this case, is always the trivial group, so the number of points in the orbit of
is
is the number of elements of
. 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.
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].
Thanks. Got ’em.
[…] 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 […]
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