Volume 8, Issue 3

For a finite group G , let k ( G ) denote the number of conjugacy classes of G . If G is a finite permutation group of degree n > 2, then k ( G ) ≤ 3 (n −1)/2 . This is an extension of a theorem of Kovács and Robinson and in turn of Riese and Schmid. If N is a normal subgroup of a completely reducible subgroup of GL( n, q ), then k ( N  ) ≤ q 5n . Similarly, if N is a normal subgroup of a primitive subgroup of S n , then k  ( N  ) ≤ p  ( n ) where p  ( n ) is the number of partitions of n . These bounds improve results of Liebeck and Pyber.

Let c S ( d  ) denote the minimal composition length of all finite solvable groups with solvable (or derived) length d . We obtain the values in the following table:

The aim of this second part is to compute explicitly the Lusztig restriction of the characteristic function of a regular unipotent class of a finite reductive group, improving slightly a theorem of Digne, Lehrer and Michel. We follow their proof but introduce new information, namely the computation of morphisms defined in the first part when v is a regular unipotent element. This new information is obtained by studying generalizations of the variety of companion matrices which were introduced by Steinberg.

Let F m be a free group with m generators and let R be a normal subgroup such that F m / R projects onto ℤ. We give a lower bound for the growth rate of the group F m / R′   (where R′   is the derived subgroup of R  ) in terms of the length ρ  = ρ ( R  ) of the shortest non-trivial relation in R . It follows that the growth rate of F m / R′   approaches 2 m  – 1 as ρ approaches infinity. This implies that the growth rate of an m -generated amenable group can be arbitrarily close to the maximum value 2 m  – 1. This answers an open question of P. de la Harpe. We prove that such groups can be found in the class of abelian-by-nilpotent groups as well as in the class of virtually metabelian groups.