On Groups of Odd Order With Exactly Two Non-Central Conjugacy Classes of Each Size

Abstract

Understanding conjugacy classes is essential to the investigation of groups. One of the first major results regarding conjugacy classes is due to Landau [6], who showed that for some given natural number r, the equation Σ= = r i 1 mi 1 1 when r m are natural numbers, has a finite number of solutions. By setting ( ( )) i G i m = o C g where i g are representatives of the conjugacy classes of a finite group G, then the class equation ( ) ( ) Σ= = r i G i 1 o C g 1 1 is such an equation. Therefore, for given r, the number of finite groups with r conjugacy classes is bounded. An interesting family of question arises: In what way does the numerical information about conjugacy classes of a finite group (the number of classes and their sizes) affect its structure? We give some examples of such results: * Burnside [2] has the following results: 1) A simple group has no conjugacy class of prime power size. 8 2) A classification of all groups G with a conjugacy class of size o(G)/2. 3) A classification of all groups with up to 5 conjugacy classes. * Poland [9] extends Burnside’s work and classifies all groups with upto 8 conjugacy classes. * In 1973, F.M.Markel [7,8] studied finite solvable groups in which distinct conjugacy classes have distinct sizes. A well known open conjecture states that 3 S is the only non-abelian finite group with conjugacy classes of distinct sizes. For solvable groups, this conjecture was proved by Zhang in [10] and independently by Knor, Lempken and Thielcke in [5]. As shown in [1], if G is a finite group, then the two conditions: (i) the conjugacy classes of G are of distinct sizes, and (ii) the noncentral conjugacy classes of G are of distinct sizes, are equivalent to each other. In this note we give a shorter proof of this result (see Corollary 3.3). If G is a finite group of odd order, then each size of a non-central conjugacy class of G appears an even number of times (see Lemma 3.1). Therefore, the corresponding problem for groups of odd order is to determine all such groups with exactly two non-central conjugacy classes of each size. The main aim of this thesis is to solve this problem. It is interesting to notice that the proof of this problem uses only very elementary results in group theory and in number theory. In particular, we do not use the famous result of Feit and Thompson [3] about the solvability of groups of odd order.