Generators and Relations
Now it’s time for the reason why free groups are so amazingly useful. Let be any set,
be the free group on
, and
be any other group. Now, every function
from
into
extends to a unique homomorphism
. Just write down any word in
, send each letter into
like the function tells you, and multiply together the result!
So what does this get us? Well, for one thing every group is (isomorphic to) a quotient of a free group. If nothing else, consider the free group
on the set of
itself. Then send each element to itself. This extends to a homomorphism
from
to
whose image is clearly all of
. Then the First Isomorphism Theorem tells us that
is isomorphic to
. That’s pretty inefficient, but it shows that we can write
like that if we want to. How can we do better?
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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.