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I have a group [itex](G,\cdot)[/itex] that has a subgroup [itex]H \leq G[/itex], and I consider the action ofHonGdefined as follows:

[itex]\varphi(h,g)=h\cdot g[/itex]

In other words, the action is simply given by the group operation.

Now I am interested in finding a (non-trivial)invariant functionw.r.t. the action ofH, which means finding a function [itex]\chi:G\rightarrow G'[/itex] such that [itex]\chi(h\cdot g)=\chi(g)[/itex] for all [itex]h\in H[/itex] and [itex]g\in G[/itex].

I realized that I can easily impose sufficient conditions on [itex]\chi[/itex] to ensure that it is an invariant function w.r.t.H.

Such conditions are:

The proof is very easy (just apply [itex]\chi[/itex] as defined in

- [itex]\chi[/itex] has the form [itex]\chi(g) = \gamma(g)^{-1}\cdot g[/itex]

- [itex]\gamma[/itex] is an
automorphismof the group [itex]G[/itex]- [itex]\gamma[/itex]
fixesthe subgroup [itex]H[/itex], [itex]\quad[/itex]i.e. [itex]\gamma(h)=h[/itex] for all [itex]h\in H[/itex]1. to [itex](h\cdot g)[/itex] and use2. and3.)

My question is: Are the above conditions already well-known, perhaps in a more general form?

Is my construction just a specific application of some more general theorem in group theory?

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# Question about invariant w.r.t. a group action

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