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

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.

• There is also partition into double cosets and further partitioning conjugacy classes by only having conjugation by elements of a subgroup. – Tobias Kildetoft Jan 24 at 13:12
• I don't know if it's "natural" or "useful" (well it's useful in at least one exercise) but you can partition $G$ as $\{\{x,x^{-1}\}, x\in G\}$ – Max Jan 24 at 13:40
• Any equivalence relation on $G$. – Shaun Jan 24 at 19:34
• Are you only interested in ways that work for all finite group or also those that work only for specific groups? – j.p. Jan 24 at 21:05
• @j.p. I'm interested primarily in those that work for all finite groups, although feel free to post about a partition that works for a particular set of groups if you think it is relevant enough (it's useful in deriving significant theorems, for example). – Leo Jan 25 at 16:48

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 $$\operatorname{Coin}(f,g) = \{x \in X \mid f(x) = g(x)\}.$$

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.