Intuition of conjugacy in symmetry groups

Question

In our course of abstract math, we were told about how to think about conjugacy in symmetry groups.

A an example, take the symmetries of a square.

• Let $a$ be the rotation by 90 degree to the left.
• Let $r$ be the reflection in the vertical line dividing the square, meaning a left to right reflection.
• Let $r^{-1} = r$, the same reflection, since $r$ is self-inverse as a reflection.

Now we want to find the conjugate $r \circ a \circ r^{-1}$.

• First, $r^{-1}$ will reflect the square.
• Then $a$ will rotate once to the left.
• Then $r$ will reflect the square back.

The overall effect is the same as having rotated the square through 270 degree to the left, call this rotation $c$.

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Now how they suggested us to think about this conjugation in a symmetry group is to imagine, in this case above, applying the final symmetry $r$ to the action of $a$. Meaning, the action of $a$ is a rotation to the left by 90 degree. Now we apply the action of $r$, a reflection from left to right to $a$. The result is a rotation by 90 degree to the right, or 270 degree to the left, which is $c$.

This makes sense to me in the specific case, but I fail to get why it is true in the general case.

I was trying to think about it in a general way:

• For a conjugate $y = g \circ x \circ g^{-1}$, we first apply $g^{-1}$, then do $x$ and then "undo" the effect of $g^{-1}$ by applying $g$.
• ... but I can't find this to be useful ...

Is there a general way to understand why the way suggested by my course makes sense?

Answer 1

Often an action (that is, a particular element of a symmetry group) can be specified uniquely by a certain feature of whatever mathematical object whose symmetry you're exploring. Moreover, the symmetries apply to these features as well. In particular, if $x$ is associated to a feature $f$, then the conjugate $gxg^{-1}$ will be associated to the feature $gf$ (that is the symmetry $g$ applied to the feature $f$). This is because $g^{-1}$ will move the feature $gf$ to $f$, then $x$ will act as it does, then $g$ moves everything back, so it's like $x$'s action was done except with respect to $gf$ instead of $f$.

(More generally, a group element may be associated to a set of features.)

This is of course vague, but you can't get any more specific than this because the concept applies so broadly to such a disparate set of situations. For change-of-bases, conjugate matrices do the same thing, but with respect to different bases. Conjugating reflections simply affects the line/plane/whatever they are across (this works in dihedral groups, orthogonal groups, affine groups, etc.) Conjugating 3D rotations simply affects the axis of rotation. Conjugating permutations affects the labels in their cycle notation (or both lines of their two-line notation). Conjugating loops in fundamental groups by paths gives loops in fundamental groups with different basepoints. (That's technically in a groupoid, not a group, but same idea.) And so on.