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Upper Bounds for the Number of Conjugacy Classes of a Finite Group

✍ Scribed by Martin W. Liebeck; László Pyber

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
315 KB
Volume
198
Category
Article
ISSN
0021-8693

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✦ Synopsis

For a finite group G, let k G denote the number of conjugacy classes of G. We prove that a simple group of Lie type of untwisted rank l over the field of q Ž . l elements has at most 6 q conjugacy classes. Using this estimate we show that for Ž . Ž .

10 n completely reducible subgroups G of GL n, q we have k G F q , confirming a Ž . conjecture of Kovacs and Robinson. For finite groups G with F* G a p-group we Ž . Ž . a a prove that k G F cp where p is the order of a Sylow p-subgroup of G and c Ž . Ž . < < X is a constant. For groups with O G s 1 we obtain that k G F G . This latter p p result confirms a conjecture of Iranzo, Navarro, and Monasor. We also improve various earlier results concerning conjugacy classes of permutation groups and linear groups. As a by-product we show that any finite group G has a soluble Ž . < < Ž . < < 3 subgroup S and a nilpotent subgroup N such that k G F S and k G F N .

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