Which ways are there to show that in any 3D point group* (a more or less complete list of those, including multiplication tables is here) the tensor product of any irreducible representation (say $\Gamma_x$) with itself always decomposes to a sum of irreducible representations that contains the trivial representation ($\Gamma_0$)?:
$\operatorname{Hom}_G(\Gamma_0, \Gamma_x \otimes \Gamma_x) = \operatorname{Hom}_G(\Gamma_0 \otimes (\Gamma_x)^*, \Gamma_x)= \operatorname{Hom}_G((\Gamma_x)^*, \Gamma_x)=\operatorname{Hom}_G(\Gamma_x, \Gamma_x)$
Any ideas, explicitly from different approaches, are appreciated. This is an attempt to get some ideas how this question could be approached. I think there might be some connection.
*) Point groups are (discrete) symmetry groups of Euclidean space which leave the origin fixed.