What natural ways of partitioning a group $G$ are there?

Question

What natural, or at least useful, ways are there to partition a finite group $G$? The two examples that come to mind are:

• Partitioning $G$ into all the left (or right) cosets of a subgroup $H$ of $G$.

• Partitioning $G$ into all its conjugacy classes.

Answer 1

One way is to generalise the notion of conjugacy to twisted conjugacy. For any endomorphism $\varphi: G \to G$ of a group $G$, you can define an equivalence relation $\sim_\varphi$ by $$g \sim_\varphi g' \iff \exists h \in G: g = hg'\varphi(h)^{-1}.$$ We call the equivalence classes $\varphi$-twisted conjugacy classes. The usual notion of conjugacy then coincides with $\sim_{\operatorname{id}}$.

This originates from topological fixed-point theory: if $f: X \to X$ is a self-map of a topological space $X$, then $f$ induces an endomorphism $f_*$ on the fundamental group $\pi_1(X)$. The number of fixed points of $f$ is related to the number of $f_*$-twisted conjugacy classes (see The theory of fixed point classes by Tsai-Han Kiang for more information).

This can be generalised even further: take two morphisms $\varphi, \psi: G \to H$. Then we can partition $H$ using the equivalence relation $\sim_\varphi^\psi$ given by $$h \sim_\varphi^\psi h' \iff \exists g \in G: h = \psi(g)h'\varphi(g)^{-1}.$$ Again, this has a topological background: if we have two maps $f,g: X \to Y$ inducing morphisms $f_*,g_*: \pi_1(X) \to \pi_1(Y)$, the number of equivalence classes is related to the number of elements of the set $$\operatorname{Coin}(f,g) = \{x \in X \mid f(x) = g(x)\}.$$

Answer 2

I suspect there may be a meta-theorem that says that (if not all, then at least most) "interesting or useful" partitions of a group $G$ arise in the following way. Take a group $H$ and let $H$ act on $G$ in some "interesting or useful" way. Then take the orbits of the action of $H$ on $G$ for your partition.

For example, the conjugacy classes of $G$ arise as the orbits of the action of the inner automorphism group of $G$ on $G$. This can be generalised by considering other subgroups of the automorphism group of $G$ (notably, the full automorphism group). If the group $G$ has additional structure (say, a group topology or an invariant measure), then one can consider just automorphisms that preserve that structure.

The cosets of a subgroup $H$ of $G$ arise as the orbits of the action of $H$ on $G$ by (left or right) multiplication.

The twisted conjugacy classes mentioned in the answer by @TastyRomeo arise from a twisted conjugacy action.

The partition mentioned by @Max in a comment comes from an action of a cyclic group of order $2$ acting by inversion on the group.