It is well-known that, if $f,g\in\mathbb{R}\to\mathbb{R}$ have compact support, then the convolution $f*g$ does too. The proof I know relies quite strongly on working over $\mathbb{R}$: compact support in $\mathbb{R}$ is equivalent to bounded support, and if $M,N$ bound $\DeclareMathOperator{supp}{supp}\supp{(f)}, \supp{(g)}$, then $M+N$ bounds $\supp{(f*g)}$.
But over any sequence space $l^p$ (or, for that matter, $c$, $c_0$, and $\mathbb{R}^{\infty}$), this property holds true, because compactness of a set is equivalent to compactness of its coordinatewise projections, by Tychonov's theorem. It's also trivially true over discrete finite groups.
Is it true over an arbitrary locally compact abelian topological group?