Studies Of Problems Associated With Conjugacy Class Sizes Of Finite Groups

In this dissertation, we investigate two problems on conjugacy class sizes of finite groups. The first one is how the maximal conjugacy class length of a finite group affects its structure. We prove that if the largest conjugacy class size of the finite group G is less than or equal to 19, then G is solvable. We observe that the largest conjugacy class size of the alternating group A5 of degree 5 is 20. Thus the upper bound 19 is best possible in order to guarantee the solvability of the corresponding group. Furthermore, it is proved that if all conjugacy class sizes of G are less than or equal to 2, then G is nilpotent. Since also the largest class size of the symmetric group S3 is 3, we get that the upper bound 2 is sharp. Note that S3 is not nilpotent. We also show that if all conjugacy class sizes of G are no more than 3, then G is supersolvable. Considering the largest class size of the nonsupersolvable group S4 is 4, the upper bound 3 is sharp. Observe that the supersolvable group< (1,2)(3,4), (1,3,4,5)> possesses conjugacy classes of the largest size 5.The second problem we discussed is how the induced conjugacy classes affect the group structure. It is an interesting phenomenon that there exist a number of analogies between results on character degrees and conjugacy class sizes in finite groups. By analysis of the definition of monomial group (for short, M-group), we give several definitions of conjugacy class monomial groups (for short, CM-groups) and strong conjugacy class monomial groups (for short, SCM-groups). Moreover, we detailed investigate the solvability and nilpotency of CM-groups and SCM-groups.