Isoclinism (reviewed)

For a group $G$ and its commutator $G'$, write $\omega_G$ for the map $G \times G \to G'$ defined by $\omega_G(x, y) = x^{-1}y^{-1}xy$, and note that $\omega_G$ descends to a map on $G / Z(G) \times G / Z(G)$. An isoclinism between two groups $G_1$ and $G_2$ is a pair of isomorphisms between their inner automorphism groups and their commutator subgroups $$\phi : G_1 / Z(G_1) \to G_2 / Z(G_2)$$ $$\sigma : G_1' \to G_2'$$ so that $$\sigma \circ \omega_{G_1} = \omega_{G_2} \circ (\phi \times \phi).$$

This is an equivalence relation on groups, and each equivalence class contains at least one stem group. Note that order is not an isoclinism invariant, but nilpotency class and solvable length are (except for the trivial group).