Finite simply reducible groups are soluble

A finite group $G$ is called simply reducible (briefly, an $SR$-group) if it has the following two properties: every element of this group is conjugate to its inverse; the tensor product of any two irreducible representations decomposes into a sum of irreducible representations of the group $G$ with multiplicities not exceeding 1. It is proved that finite $SR$-groups are soluble.
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