tailieunhanh - Groups with the given set of the lengths of conjugacy classes

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 .