Compatible pair of actions

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Definition

Definition with left action convention

Suppose G and H are groups. Suppose \alpha:G \to \operatorname{Aut}(H) is a homomorphism of groups, defining a group action of G on H. Suppose \beta:H \to \operatorname{Aut}(G) is a homomorphism of groups, defining a group action of H on G. For g \in G, denote by c_g: G \to G the conjugation map by g. See group acts as automorphisms by conjugation. Then, we say that the actions \alpha,\beta form a compatible pair if both these conditions hold:

  • \beta(\alpha(g_1)(h))(g_2) = c_{g_1}(\beta(h)(c_{g_1^{-1}}(g_2)))) \ \forall \ g_1,g_2 \in G, h \in H
  • \alpha(\beta(h_1)(g))(h_2) = c_{h_1}(\alpha(g)(c_{h_1^{-1}}(h_2))) \ \forall \ h_1,h_2 \in H, g \in G

The above expressions are easier to write down if we use \cdot to denote all the actions. In that case, the conditions read:

  • (g_1 \cdot h) \cdot g_2 = g_1 \cdot (h \cdot (g_1^{-1} \cdot g_2)) \ \forall \ g_1,g_2 \in G, h \in H
  • (h_1 \cdot g) \cdot h_2 = h_1 \cdot (g \cdot (h_1^{-1} \cdot h_2)) \ \forall \ h_1,h_2 \in H, g \in G

Here is an equivalent formulation of these two conditions that is more convenient:

  • c_{g_1}(\beta(h)g_2) = \beta(\alpha(g_1)h)(c_{g_1}(g_2)) \ \forall \ g_1,g_2 \in G, h \in H (the g_2 here is the c_{g_1^{-1}}g_2 of the preceding formulation).
  • c_{h_1}(\alpha(g)h_2) = \alpha(\beta(h_1)g)(c_{h_1}(h_2)) \ \forall \ g \in G, h_1, h_2 \in H (the h_2 here is the c_{h_1^{-1}}h_2 of the preceding formulation)

In the \cdot notation, these become:

  • g_1 \cdot (h \cdot g_2) = (g_1 \cdot h) \cdot (g_1 \cdot g_2) \ \forall \ g_1,g_2 \in G, h \in H (the g_2 here is the g_1^{-1} \cdot g_2 of the preceding formulation).
  • h_1 \cdot (g \cdot h_2) = (h_1 \cdot g) \cdot (h_1 \cdot h_2) \ \forall g \in G, h_1,h_2 \in H (the h_2 here is the h_1^{-1} \cdot h_2 of the preceding formulation)

Definition with right action convention

We can give a corresponding definition using the right action convention, but the literature uses the left action convention, so this definition is intended purely as an illustrative exercise. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Symmetry in the definition

Given groups G and H with actions \alpha:G \to \operatorname{Aut}(H) and \beta:H \to \operatorname{Aut}(G), \alpha is compatible with \beta if and only if \beta is compatible with \alpha. In other words, the definition of compatibility is symmetric under interchanging the roles of the two groups.

Particular cases