We classify the real and strongly real conjugacy classes in PGLnðqÞ, PSLnðqÞ, and all quasi-simple covers of PSLnðqÞ. In each case we give a formula for the number of real, and the number of strongly real, conjugacy classes. This is a companion paper to [4] in which we classified the real and strongly real conjugacy classes in GLnðqÞ and SLnðqÞ.

We classify the real and strongly real conjugacy classes in PGLnðqÞ, PSLnðqÞ, and all quasi-simple covers of PSLnðqÞ. In each case we give a formula for the number of real, and the number of strongly real, conjugacy classes. This is a companion paper to [4] in which we classified the real and strongly real conjugacy classes in GLnðqÞ and SLnðqÞ.

INTRODUCTION AND NOTATION: As usual, for G a finite group, we denote the set of non-identity elements of G by G, the number of conjugacy classes of G by k(G) and the set of complex irreducible characters ofG by Irr(G). When H is a non-trivial subgroup of G, we let k+(G,H) denote the number of irreducible characters of G which do not vanish identically on H (and we remark that k+(G,H) = k(G) whe...

The subgroup pattern of a finite groups G is the table of marks of G together with a list of representatives of the conjugacy classes of subgroups of G. In this article we present an algorithm for the computation of the subgroup pattern of a cyclic extension of G from the subgroup pattern of G. Repeated application of this algorithm yields an algorithm for the computation of the table of marks ...

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....

To classify the finite dimensional pointed Hopf algebras with Weyl group G of E6, E7, F4, G2, we obtain the representatives of conjugacy classes of G and all character tables of centralizers of these representatives by means of software GAP. 2000 Mathematics Subject Classification: 16W30, 68M07 keywords: GAP, Hopf algebra, Weyl group, character.

To classify the finite dimensional pointed Hopf algebras with Weyl group G of E6, E7, F4, G2, we obtain the representatives of conjugacy classes of G and all character tables of centralizers of these representatives by means of software GAP. 2000 Mathematics Subject Classification: 16W30, 68M07 keywords: GAP, Hopf algebra, Weyl group, character.

To classify the finite dimensional pointed Hopf algebras with Weyl group G of E8, we obtain the representatives of conjugacy classes of G and all character tables of centralizers of these representatives by means of software GAP. In this paper we only list character table 65–94. 2000 Mathematics Subject Classification: 16W30, 68M07 keywords: GAP, Hopf algebra, Weyl group, character.