In why this !!\sigma\pi\sigma^{-1}!! keeps apearing in my group
theory book? (cycle
decomposition) the
querent asked about the “conjugation” operation that keeps cropping
up in group theory. Why is it important? I sympathize with this;
it wasn't adequately explained when I took group theory, and I had
to figure it out a long time later. Unfortunately I don't think I
picked the right example to explain it, so I am going to try again
now.
Consider the eight symmetries of the square. They are of five types:
- Rotation clockwise or counterclockwise by 90°.
- Rotation by 180°.
- Horizontal or vertical reflection
- Diagonal reflection
- The trivial (identity) symmetry
What is meant when I say that a horizontal and a vertical reflection
are of the same ‘type’? Informally, it is that the horizontal
reflection looks just like the vertical reflection, if you turn your
head ninety degrees. We can formalize this by observing that if we
rotate the square 90°, then give it a horizontal flip, then rotate it
back, the effect is exactly to give it a vertical flip. In notation,
we might represent the horizontal flip by !!H!!, the vertical flip by
!!V!!, the clockwise rotation by !!\rho!!, and the counterclockwise
rotation by !!\rho^{-1}!!; then we have
$$ \rho H \rho^{-1} = V$$
and similarly
$$ \rho V \rho^{-1} = H.$$
Vertical flips do not look like diagonal flips—the diagonal flip leaves two of the corners in the same place, and the vertical flip does not—and indeed there is
no analogous formula with !!H!! replaced with one of the diagonal
flips. However, if !!D_1!! and !!D_2!! are the two diagonal flips,
then we do have
$$ \rho D_1 \rho^{-1} = D_2.$$
In general, When !!a!! and !!b!! are
two symmetries, and there is some symmetry !!x!! for which
$$xax^{-1} = b$$
we say that !!a!! is conjugate to !!b!!.
One can show that
conjugacy is an equivalence relation, which means that the symmetries
of any object can be divided into separate “conjugacy classes” such that two
symmetries are conjugate if and only if they are in the same class.
For the square, the conjugacy classes are the five I listed earlier.
This conjugacy thing is important for telling when two symmetries
are group-theoretically “the same”, and have the same
group-theoretic properties. For example, the fact that the
horizontal and vertical flips move all four vertices, while the
diagonal flips do not. Another example is that a horizontal flip is
self-inverse (if you do it again, it cancels itself out), but a 90°
rotation is not (you have to do it four times before it cancels
out.) But the horizontal flip shares all its properties with the
vertical flip, because it is the same if you just turn your head.
Identifying this sameness makes certain kinds of arguments much
simpler. For example, in counting
squares, I wanted to
count the number of ways of coloring the faces of a cube, and instead
of dealing with the 24 symmetries of the cube, I only needed to deal
with their 5 conjugacy classes.
The example I gave in my math.se answer was maybe less perspicuous. I
considered the symmetries of a sphere, and talked about how two
rotations of the sphere by 17° are conjugate, regardless of what axis
one rotates around. I thought of the square at the end, and threw it
in, but I wish I had started with it.