I've been working on a question for a few days now, and I'm stuck on proving a claim that I don't know if there's any reason for it to be true. I'll write it here in the greatest generality I can think of, in hopes that someone might know to tell me if they see any reason for this to be true or untrue:\
Prove or Find a counterexample:
Let $G$ be a (perhaps infinite) group, and let $H$ be an abelian group. Suppose that $G$ acts on $H$ in a way such that the $G$-orbits in $H$ are finite. Let $a,b\in H$ be arbitrary. Is it true that $$\left|G_{a+b}\right|\le\max\lbrace\left|G_a\right|,\left|G_b\right|\rbrace$$ (where $G_x$ denotes the orbit of an element $x\in H$)?
I will be very grateful to anyone who can shed some light on this claim, or maybe suggest extra condition underwhich this might hold.
Thank you.
For an easy counterexample, let $G=H=\mathbb{Z}$, and let $G$ act on $H$ by fixing the elements $1,2\in H$, and let $g\in G$ swap the other elements of $H$ in pairs $\{-3,-2\},\;\{-1,0\},\;\{3,4\},\ldots$ or fix them according to the parity of the element of $g$. Then $$G_1=\{1\},\quad G_2=\{2\},\quad G_3=\{3,4\}.$$ Note that the group structure of $H$ was irrelevant. For all we cared, as long as $a+b$ was different than $a$ and $b$, we could make the action do whatever we wanted differently to them.
I suppose it's possible you could make the statement true if there's supposed to be some interaction between the operations of $G$ and $H$, though I can't think of anything specific that would work.