Why do elements of the same conjugacy class tend to have similar properties? I am trying to develop an intuitive understanding of why elements of the same conjugacy class tend to have similar properties. For example, the rotations form a conjugacy class in the dihedral groups, and so do the flips. It is clear in some particular cases why they have similar properties, but a general understanding eludes me. Similar matrices, for example, have similar properties because they are really the same matrix but under a different basis. I considered that it may be because elements of the same conjugacy class must have the same order, but I'm not sure this is enough, since elements of the same order do not always have similar properties. 
 A: It's because there's an automorphism of the group taking one to the other.
If $y=gxg^{-1}$ in a group $G$, then $z\mapsto gzg^{-1}$ is an automorphism of $G$ taking $x$ to $y$ (an inner automorphism).
A: Most instances of conjugation can be considered a "change of basis" transformation, analogous to what you're familiar with from linear algebra. 
For an example of permutations in the symmetric group, if $\sigma = (1\ 3\ 4)$ and $\tau = (1\ 5\ 2\ 3)$, then $\sigma^{-1}\tau\sigma = (\sigma(1)\  \sigma(5)\ \sigma(2)\ \sigma(3)) = (3\ 5\ 2\ 1)$; essentially, you relabel the elements $\{1,2,3,4\}$ being acted upon according to $\tau$, perform the permutation $\sigma$, and then switch back to the original labeling using $\tau^{-1}$, exactly analogous to the change of basis situation in linear algebra. In the dihedral groups, a similar thing happens relabeling (and "unrelabeling") the vertices of the $n$-gon -- provided you use some element of the dihedral group to do the relabeling!
Since any group can be thought of as a permutation group (i.e., a subgroup of some symmetric group), this sort of thing happens in general. In fact, permutations can be represented as permutation matrices, so we can represent groups using groups of matrices, as well. Then conjugation is the familiar change of base transformation from linear algebra, with the caveat that we can only conjugate by matrices that correspond to group elements (like relabeling vertices of the $n$-gon, in the dihedral group case), not just any matrix.
