Intuition for Conjugacy Classes in Groups On the Wikipedia page for Conjugacy Classes, it says that "members of the same conjugacy class cannot be distinguished using the group structure alone".
In what sense is this true?
I can see that this is not meant to taken literally: for example, it's not true that if two elements of a group are conjugate, they act the same way in the group multiplication table.  So in what sense can conjugate elements "not be distinguished"? The best that I can come up with is that conjugate elements have the same order -- but the opposite direction is not even true in general! (i.e: $|x| = |y|$ does not imply $x$ and $y$ are conjugate) Are there other properties conjugate elements share -- properties that $x$ and $y$ share if and only if they are conjugate?
To be clear, I understand why conjugacy is important in certain examples. For instance:

*

*In $GL(n, \mathbb{F})$, for example, if two matrices are conjugate then they have the same rank, nullity, trace, determinant, and so on.

*In $S_n$, two permutations are conjugate if and only if they have the same cycle type.

But in a general group $G$, I cannot see why saying "$x$ and $y$ are conjugate" is significant / what it tells us. Any clarification would be much appreciated. Thanks!
 A: Conjugate elements of a group, and for that matter elements that are conjugate by an automorphism, share all "purely group-theoretic properties." Some examples:

*

*The order of the element $\text{ord}(g) = | \langle g \rangle |$, where $\langle g \rangle$ denotes the subgroup generated by $g$

*The number of $k^{th}$ roots $\{ h \in G : h^k = g \}$, for any $k$ (and in fact more than this, e.g. the conjugacy-by-automorphism class)

*The number of ways to write $g$ as a commutator $[h, k] = hkh^{-1}k^{-1}$ (this and the previous example generalize to counting solutions to systems of equations in $G$ with $g$ as a parameter)

*The isomorphism class of the centralizer $C_G(g) = \{ h \in G : hg = gh \}$

*The isomorphism class of the normalizer $N_G( \langle g \rangle ) = \{ h \in G : h \langle g \rangle = \langle g \rangle h \}$

*Various combinations of the above constructions

Elements which are conjugate (by an inner automorphism) and not just conjugate by an automorphism share a few more properties:

*

*The centralizer (not just up to isomorphism but on the nose)

*The conjugacy class of the image under any group homomorphism $f : G \to H$ (e.g. a permutation representation $G \to S_n$, or a linear representation $G \to GL_n$; so conjugate elements have the same cycle type with respect to any permutation action and the same eigenvalues etc. with respect to any linear representation)

*The value when evaluated on any character (this is an if-and-only-if for finite groups: two elements $g, h$ of a finite group are conjugate iff $\chi(g) = \chi(h)$ for every irreducible character $\chi$ over $\mathbb{C}$).

