Let G be a finite group and let g∈G. In 1896, Frobenius showed that the number of ways to express g as a commutator of elements of G is |G|FG(g), where FG(g)=∑χ∈ Irr (G)χ(g)/χ(1) is the Frobenius character sum. This sum received particular attention in the case where G is a (non-abelian) finite simple group, and some related conjectures were posed. In this paper we discuss these conjectures, refute one of them, and provide partial evidence in favor of another one.
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