Some applications of graph theory to finite groups

AbstractResults on vertex coloring and the vertex independence number of a finite graph are used to prove:Theorem. Let G be a finite group with conjugacy classes indexed by cardinality: 1 = |[x1]| ⩽|[x2]| ⩽···, and let CG(x) denote the centralizer of x. If m is the smallest integer i such that |[x1]|+|[x2]|+···+|[x1]|⩾|C(x1)|, then each abelian subgroup A of G has card inality|A|⩽ |[x1]|+|[x2]|+···+|[xm]|.Theorem. Let G be a finite group with a proper subgroup M, suchthat x∈M−{1}⇒CG(x)⊆ M. Then G contains at least [|G|13] pairwise non-commuting elements, and hence G cannot be covered by the union of fewer than [|G|13] abelian subgroups.Theorem. Let S be a locally maximal sum-free subset of the abelian group G. Then |S−S|+|SU−S|−3⩽|G|(1−|S−S|-1), with equality if and only if S−S is a subgroup H of G, [G:H]=3, and S is a coset of H.Some open problems are also stated