Some applications of graph theory to finite groups

Abstract

Results 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 = |[ x 1 ]| ⩽|[ x 2]| ⩽···, and let C G ( x) denote the centralizer of x. If m is the smallest integer i such that |[ x 1]|+|[ x 2]|+···+|[ x 1]|⩾| C( x 1)|, then each abelian subgroup A of G has card inality|A|⩽ |[ x 1]|+|[ x 2]|+···+|[ x m]|. Theorem. Let G be a finite group with a proper subgroup M, suchthat x∈ M−{1}⇒ C G( x)⊆ M. Then G contains at least [| G| 1 3 ] pairwise non-commuting elements, and hence G cannot be covered by the union of fewer than [| G| 1 3 ] 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.