[MSE] Help on a special kind of functional equation.

[JmsNxn]

Is this some kind of semi-direct product? Jesus; that would be cool...

Let me think on this... I'll see if I can think of anything.

Is \( g^\alpha \) really supposed to be an element of \( \text{Aut} \); rather than necessarily a conjugation?

........

Update:

So I'm not certain I can answer this; but, did you understand when I introduced the congruent integral (if you read the paper)? This is a different form of the compositional integral that is in a "modded out space". I can reduce (I think); your question into the abelian case; but I need to know how much of the congruent integral I should explain.

Also; what the solutions \( f \) are referred to as is homormorphisms of the semi-direct products between two groups.

In this case you are taking an inner automorphism \( g^{\alpha}\in \text{InnAut}(G) \subset \text{Aut}(G) \). Then you are constructing what is typically written,

\(
G \propto_{g^\alpha} N\\
\)

And you are looking for

\(
f : G \propto_{g^\alpha} N \to G
\)

There's a word for this; I can't remember it exactly.

Also, this forum doesn't have the best latex implementation; so \( \propto \) should actually be the symbol \( \rtimes \); which is more angular.  It should look more like this 
.....

I'll explain this better tomorrow. Long night; but it's at least SOMETHING like this. Nonetheless; your answer lies in semi-direct products.