Conjugacy in groups?

Question

Regarding, conjugate classes I am confused and need help. It is one of the below, or the both:

i) It is w.r.t. the object (group) and the configuration achieved; or
ii) it is for each individual action and the corresponding permutation.

I feel that it incorporates both (i) & (ii) above, i.e. it considers for making one set of symmetry (actions) to be equivalent to the other iff equivalence given by (ii) is satisfied, while one can substitute a given set of symmetry (actions) with a differing one if the actions result in same configuration of the object (or (i)).

My confusion arises from here on MSE that the set of diagonal reflections are different from edge reflections, for the case of even order of the group. $D_4$ has order in terms of number of symmetries (actions) as $8$, an even number and so should have difference between the two types of reflections.

=> So does the answer mean that the diagonal reflections are equivalent to each other?
I am listing the permutations for $D_4$, induced by $R_{90}, D_{1,3}, D_{2,4}, D_x, D_y$, where $D_{1,3} =$ reflection across the diagonal joining vertices $1,3$; while for $D_y$ it is the reflection across the y-axis.

$R_{90} = (1 4 3 2)$

$D_{1,3} = (24)(1)(3) $

$D_{2,4} = (13)(2)(4)$

$D_x = (14)(23)$

$D_y = (12)(34)$

But, all the 4 reflections are equivalent in one respect that need only two operations of the set of rotations, $R_i, 0\le i\le 3$ & any other single reflection to describe all other operations in terms of them.

So, how can I understand the concept of conjugacy is not clear to me.

But have a doubt in my understanding of the above. It talks about order of $D_n$, as : "the dihedral group of order $n$, the set of symmetries of a regular $n-$gon". I feel that the order is : $8=(4+4)$, as there are 4 symmetries of reflections and 4 symmetries of rotations (including the identity one). This also stated [at this MSE question][2]. If so, then how it is satisfying the criteria of conjugacy? Am I interpreting the conjugacy wrong? My interpretation is that it refers to the final configuration of the vertices w.r.t. each other. From that respect, all four reflections produce the same orientation of the 4 vertices. I do not see it as in terms of individual actions' permutations, and hence differing permutations.

Answer 1

$D_4$ is, as you say, the group of euclidean symmetries of a square. One has to distinguish between this group of maps of the plane and the group of induced permutations of the vertices, which is a subgroup $D_4'$ of the symmetric group $S_4:={\cal S}\bigl(\{1,2,3,4\}\bigr)$.

In $D_4$ we have the $4$ rotations by $0^\circ$, $90^\circ$, $180^\circ$, and $-90^\circ$; all of these modulo $360^\circ$. Then there are two reflections across the two diagonals, and finally there are two reflections across symmetry axes parallel to the sides of the square.

$R_{180}$ and the reflections are of order $2$, hence their own inverses; furthermore $R_{-90}=R_{90}^{-1}$. By going through the motions one easily verifies that $D_{13}\circ R_{90}\circ D_{13}=R_{-90}$, hence $R_{90}$ and $R_{-90}$ are conjugate. Similarly $R_{-90}\circ D_{13}\circ R_{90}=D_{24}$ and $R_{-90}\circ D_x\circ R_{90}=D_y$. Finally $R_{180}$ is only conjugate to itself.

These "geometric" maps induce permutations of the vertices, which then can be studied in terms of cycle structure. Concerning conjugation one can say the following: Assume that $\pi$ is a permutation of some finite set $S$, and use $\sigma\in{\cal S}(S)$ for conjugating $\pi$ to another permutation $\pi':=\sigma\circ \pi\circ\sigma^{-1}$. If $\pi(a)=b$ then $\pi'\bigl(\sigma(a)\bigr)=\sigma(b)$. If in such a case $(a_1,a_2,\ldots a_r)$ is a cycle of $\pi$ then $\bigl(\sigma(a_1),\ldots,\sigma(a_r)\bigr)$ is a cycle of $\pi'$. It follows that $\pi$and $\pi'$ have the same cycle spectrum. If the full symmetric group is at stake then this argument can be reversed: If $\pi$ and $\pi'$ have the same cycle spectrum then they are conjugate in ${\cal S}(S)$.

In our example the permutation of $[4]$ induced by $R_{180}$ has the cycle decomposition $(13)(24)$, and the permutation induced by $D_x$ has the cycle decomposition $(14)(23)$. By the foregoing these two permutations are conjugate in $S_4$. But $R_{180}$ and $D_x$ are not conjugate in $D_4$; the reason being that $D_4'$ is only a proper subgroup of $S_4$, and the $\sigma\in S_4$ used to conjugate between $(13)(24)$ and $(14)(23)$ does not lie in $D_4'$.