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Primary and biprimary class sizes implying nilpotency of finite groups

Let G be a finite group. We prove that G is nilpotent if the set of conjugacy class sizes of primary and bipirimary elements is {1, m, n, mn} with m and n coprime. Moreover, m and n are distinct primes power. | Turk J Math (2016) 40: 389 – 396 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1504-79 Research Article Primary and biprimary class sizes implying nilpotency of finite groups Qinhui JIANG∗, Changguo SHAO School of Mathematical Sciences, University of Jinan, Shandong, P.R. China Received: 26.04.2015 • Accepted/Published Online: 24.08.2015 • Final Version: 10.02.2016 Abstract: Let G be a finite group. We prove that G is nilpotent if the set of conjugacy class sizes of primary and bipirimary elements is {1, m, n, mn} with m and n coprime. Moreover, m and n are distinct primes power. Key words: Finite groups, conjugacy class sizes, primary and biprimary elements 1. Introduction Throughout this paper all groups considered are finite and G always denotes a group. For an element x of a group G we denote by xG the conjugacy class containing x, and by |xG | the conjugacy class size of x. A primary element is an element of prime power order and a biprimary (triprimary) element is an element whose order is divisible by precisely two (three) primes. The rest of the notation and terminology is standard; readers may refer to [7]. In recent years, there has been tremendous interest in studying the structure of a group by some arithmetical conditions imposed on the conjugacy class sizes of G . A classical result due to Itˆo [8] is that a group G with two conjugacy class sizes is nilpotent and G is solvable if it has three conjugacy class sizes. Beltr´an and Felipe [3, 2] studied groups with four conjugacy class sizes and proved that if the set of conjugacy class sizes of G is {1, m, n, mn} with integers m, n > 1 coprime, then G is nilpotent with m and n distinct primes power. To investigate the influence of partial conjugacy class sizes on the structure of groups is also an interesting topic. For instance, Li [11] proved that a group G is solvable if its conjugacy class size of every primary element is either 1 or m