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2019•Communications in AlgebraRequires access

Determining group structure by set of conjugacy class sizes

Changguo Shao, Qinhui Jiang

Open publisher page 4 citations

Abstract

Let G be a finite group. Let further m1,m2,m3 be three positive integers such that m1 and m2 do not divide each other and m1m2 is coprime to m3. We prove that if the set of conjugacy class sizes of a finite group G is {1,m1,m2}×{1,m3}, then G=A×B, where A and B are satisfying cs(A)={1,m1,m2} and cs(B)={1,m3} with m3 a prime power. In particular, G is solvable.

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What this paper is about

Let G be a finite group. Let further m1,m2,m3 be three positive integers such that m1 and m2 do not divide each other and m1m2 is coprime to m3. We prove that if the set of conjugacy class sizes of a finite group G is {1,m1,m2}×{1,m3}, then G=A×B, where A and B are satisfying cs(A)={1,m1,m2} and cs(B)={1,m3} with m3 a prime power. In particular, G is solvable.

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Available abstract

Let G be a finite group. Let further m1,m2,m3 be three positive integers such that m1 and m2 do not divide each other and m1m2 is coprime to m3. We prove that if the set of conjugacy class sizes of a finite group G is {1,m1,m2}×{1,m3}, then G=A×B, where A and B are satisfying cs(A)={1,m1,m2} and cs(B)={1,m3} with m3 a prime power. In particular, G is solvable.

Key concepts: Conjugacy class, Coprime integers, Mathematics, Finite group, Class (philosophy), Prime (order theory), Set (abstract data type), Group (periodic table)

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