On tensor squares of irreducible representations of almost simple groups. I

Almost simple $\mathrm{SM}_m$-groups are considered. A group $G$ is called a $\mathrm{SM}_m$-group if the tensor square of any irreducible representation is decomposed into the sum of its irreducible representations with multiplicities not greater than $m$. In the first part of this article we consider simple groups. It turned out that among them only groups $L_2(q)$, $q=2^t$, $t>1$, are $\mathrm{SM}_2$-groups.