Thompson's conjecture on determination of simple group by its conjugacy class size set

Statement
If $$G,M$$ are finite groups, such that the conjugacy class size set of $$G$$ is the same as that of $$M$$, and such that $$G$$ is a fact about::finite simple non-abelian group while $$M$$ is a fact about::centerless group, then $$G$$ and $$M$$ are isomorphic groups.