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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 a unique stem group. Note that order is not an isoclinism invariant, but nilpotency class and solvable length are (except for the trivial group).

Knowl status:
  • Review status: reviewed
  • Last edited by Jennifer Paulhus on 2022-07-19 09:15:57
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