While thinking about item (2) in [Standard or good names for relations between maps](https://mathoverflow.net/q/437261), I thought I'd look at the relation $x \sim g x g$ in groups.


Going through all small groups of order at most 64, it seems to me, that for any finite group the connected components all have the same size, and that the number of connected components is a power of 2. Is this obvious, or false?


NB: What I am really after is a name for the relation.


(edited to reflect YCor's observation)


(edited to reflect LSpice's observation)



The relation $x \sim g x g$ in groups
It's indeed quite immediate.


Indeed, let $\simeq$ be the equivalence relation generated by this relation. Then $x\simeq y$ iff the images of $x$ and $y$ in $G/G^2$ are equal. Here $G^2$ is the subgroup of $G$ generated by squares, so $G/G^2$ is the largest 2-elementary abelian quotient of $G$. ($G$ is not assumed finite.)


To prove the claim, one direction is clear ($x\simeq y$ implies $xG^2=yG^2$). For the converse, first observe that
$$x\simeq gxg\simeq hgxgh\simeq (hg)^{-1}hgxgh(hg)^{-1}=xghg^{-1}h^{-1},$$
and by induction it follows that $x\simeq xz$ for every $z\in [G,G]$. Then
$$xg^2=gxg\; g^{-1}(x^{-1}g^{-1}xg)g\simeq gxg\simeq x.$$
So by induction $xz\simeq x$ for every $z\in G^2$.


(For a name: it's the equivalence relation induced by equality modulo a normal subgroup, or equivalently, fibers of a group homomorphism, so this is the most standard kind of equivalence relations in group theory.)


