Remark 3

One can prove (cf. R. Guralnick and P.H. Tiep, Decomposition of small tensor powers and Larsen’s conjecture, Representation Theory 9 (2005), 138–208) that the condition that G be infinite can be omitted if one assumes that E is of dimension $\geqslant 7$ and if one replaces the hypothesis $M_4(\pi ) = 2$ (resp. $M_4(\pi ) = 3$) by $M_8(\pi ) = 24$ (resp. $M_8(\pi ) = 105$).