We study the structures of some finite groups such that the conjugacy class size of every noncentral element of them is divisible by a prime p. | Turk J Math (2015) 39: 507 – 514 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Groups with the given set of the lengths of conjugacy classes Neda AHANJIDEH∗ Department of Pure Mathematics, Faculty of Mathematical Sciences, Shahrekord University, Shahrekord, Iran Received: • Accepted/Published Online: • Printed: Abstract: We study the structures of some finite groups such that the conjugacy class size of every noncentral element of them is divisible by a prime p . Key words: Conjugacy class sizes, F-groups 1. Introduction Let G be a finite group and Z(G) be its center. For x ∈ G , suppose that clG (x) denotes the conjugacy class in G containing x and CG (x) denotes the centralizer of x in G . We will use cs(G) for the set {n : G has a conjugacy class of size n} . It is known that some results on character degrees of finite groups and their conjugacy class sizes are parallel. Thompson in 1970 (see [6]) proved that if the degree of every nonlinear irreducible character of the finite group G is divisible by a prime p, then G has a normal p-complement. Along with this question, Caminas posed the following question: Question. [1, Question 8.] If the conjugacy class size of every noncentral element of a group G is divisible by a prime p , what can be said about G ? It is known that cs(GL2 (q n )) = {1, q 2n − 1, q n (q n + 1), q n (q n − 1)}. Thus, if q is an odd prime, then cs(GL2 (q n )) = {1, , 2e2 .n2 , 2e3 .n3 }, where 1 n2 > n3 are odd natural numbers. This example shows the existence of the finite groups where the conjugacy class size of their noncentral elements is divisible by a prime p but contains no normal p -complements. Thus, Thompson’s result and the answer to the above question are not necessarily parallel. This example motivates us to find the structure of the finite group G with cs(G) = {1, pe1 n1 , pe2 n2 , . . . , pek nk .
