Let $G$ and $H$ be groups, both acting on a set $X$. Suppose that there is a homomorphism $\phi:G\to H$ such that for every $g\in G$ and $x\in X$, $g\cdot x = \phi(g)\cdot x$. Is there a name for this particular situation? What about in the case that $\phi$ is a surjection/quotient map? The latter is a particular case of orbit equivalence of actions, but seems to deserve a stronger name.
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1$\begingroup$ Note that ${\rm ker} \phi$ acts trivially on $X.$ Hence the $G$-set $X$ is the inflation of $X$ as ${\rm Im} \phi$-set. $\endgroup$– Geoff RobinsonCommented Jul 14, 2014 at 20:51
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$\begingroup$ What is the definition of "inflation" here? $\endgroup$– Iian SmytheCommented Jul 14, 2014 at 21:06
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1$\begingroup$ It is more usually used for modules. If $N \lhd G$ and $G/N$ acts on some structure, "inflation" to an action of $G$ is just the process of making $G$ act on the same structure by letting $N$ acts trivially, and letting $g$ act as the coset $gN$ did for each $g \in G$ - the point being that this action is well-defined. $\endgroup$– Geoff RobinsonCommented Jul 14, 2014 at 21:22
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1$\begingroup$ I would say that the identity map of $X$ is equivariant with respect to the two action of $G$. $\endgroup$– user43326Commented Jul 15, 2014 at 8:39
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2$\begingroup$ How about "$\phi$ is action preserving"? Or "$\phi$ is an action homomorphism"? Underlying this is the observation that $\phi$ is a morphism in a category of group actions on $X$. $\endgroup$– Lee MosherCommented Jul 21, 2014 at 17:49
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