Attempt 3 questions. (If you attempt more, only the best 3 will be counted.) All questions carry the same number of marks. Unless otherwise stated, all groups are compact (or finite), and all representations are of finite degree over C. 1. Define a group representation. What is meant by saying that a representation α is simple? Determine all simple representations of S 3. from first principles....

The problem of enumeration of conjugacy classes of finite abelian subgroups of the mapping class group Mσ of a compact closed surface X of genus σ is considered. A complete method of enumeration is achieved for finite elementary abelian subgroups and steps are taken toward enumera-tion of finite abelian subgroups.

Denote k(G) the number of conjugacy classes of a group G. Some inequalities are deduced by arithmetic means for k(G), where G is a p-group. As an application, k(G) is calculated for special cases of p-groups. A method of estimating k(G) for some finite groups, others then p-groups is also presented.