Executive Summary : | Wreath product of groups, first introduced by Alfred Loewy in 1927, is an essential tool in group theory. It is used to study various important groups, such as Weyl groups, cyclic groups of order 2, and symmetric groups. Centralizers of elements in symmetric groups are direct products of wreath products of groups. Two elements in a group G are called z-equivalent if their centralizers are conjugate subgroups within G. These equivalence classes are called z-classes, which are useful for enumerating the number of simultaneous conjugacy classes of commuting tuples of elements in groups. The researchers are interested in studying z-classes in wreath products of groups, starting with generalized symmetric groups. They have also determined that for alternating groups, the number of rational conjugacy classes is the same as the number of rational valued characters. If an element g in a finite group G is rational, all its conjugate elements are rational, and its conjugacy class is said to be rational. The researchers plan to study rational conjugacy classes and rational valued characters for other simple groups and finite simple groups. They use computer algebra software GAP to study this problem for groups of small order, finding encouraging results. A group is considered a rational group if all elements are rational. However, all generalized symmetric groups are not rational. The researchers aim to determine the rational conjugacy classes, rational valued irreducible characters, and study the relation between these two. They are also interested in studying word maps and their images for certain classes of p-groups. |