An extension of a theorem of Alan Camina on conjugacy class sizes

作者：Shao Changguo; Jiang Qinhui^*

来源：Israel Journal of Mathematics, 2014, 204(1): 145-153.

DOI：10.1007/s11856-014-1079-y

Let G be a finite group. Let n be a positive integer and p a prime coprime to n. In this paper we prove that if the set of conjugacy class sizes of primary and biprimary elements of group G is {1,p (a) , p (a) n}, then G a parts per thousand OE G (0) x H, where H is abelian and G (0) contains a normal subgroup M x P (0) of index p. Moreover, M x P (0) is the set of all elements of G (0) of conjugacy class sizes p (a) or 1, where M is an abelian pi(n)-subgroup of G (0) and P (0) is an abelian p-subgroup of G (0), neither being central in G. Finally, p (a) = p and P/P (0) acts fixed-point-freely on M and I center dot(P) a parts per thousand currency sign Z(P). This is an extension of Alan Camina's theorems on the structure of groups whose set of conjugacy class size is {1,p (a) , p (a) q (b) }, where p and q are two distinct primes.

• 出版日期2014-10
• 单位济南大学