Compatible pair of actions

Definition

Definition with left action convention

Suppose $G$ and $H$ are groups. Suppose $\alpha :G\to \mathrm{A}\mathrm{u}\mathrm{t}(H)$ is a homomorphism of groups, defining a group action of $G$ on $H$. Suppose $\beta :H\to \mathrm{A}\mathrm{u}\mathrm{t}(G)$ 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:

• $\beta (\alpha (g_1)(h))(g_2)=c_{g_1}(\beta (h)(c_{g_1^{-1}}(g_2))))\mathrm{\forall }{g}_1,g_2\in G,h\in H$
• $\alpha (\beta (h_1)(g))(h_2)=c_{h_1}(\alpha (g)(c_{h_1^{-1}}(h_2)))\mathrm{\forall }{h}_1,h_2\in H,g\in G$

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

• $(g_1\cdot h)\cdot g_2=g_1\cdot (h\cdot (g_1^{-1}\cdot g_2))\mathrm{\forall }{g}_1,g_2\in G,h\in H$
• $(h_1\cdot g)\cdot h_2=h_1\cdot (g\cdot (h_1^{-1}\cdot h_2))\mathrm{\forall }{h}_1,h_2\in H,g\in G$

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

• $g_1\cdot (h\cdot g_2)=(g_1\cdot h)\cdot (g_1\cdot g_2)\mathrm{\forall }{g}_1,g_2\in G,h\in H$ (the $g_2$ here is the $g_1^{-1}\cdot g_2$ of the preceding formulation).
• $h_1\cdot (g\cdot h_2)=(h_1\cdot g)\cdot (h_1\cdot h_2)\mathrm{\forall }g\in G,h_1,h_2\in H$ (the $h_2$ here is the $h_1^{-1}\cdot h_2$ of the preceding formulation)

Facts

• Compatible with trivial action iff trivial action

Particular cases

• If both the actions are trivial, i.e., both the homomorphisms $\alpha ,\beta$ are trivial maps, then they form a compatible pair.
• If $G,H$ are both subgroups of some group $Q$ that normalize each other (i.e., each is contained in the normalizer of the other), and $\alpha ,\beta$ are the actions of the groups on each other by conjugation, then they form a compatible pair. Note that in this case, all the actions are just conjugation in $Q$ and checking the conditions simply amounts to checking two words to be equal.