On the number of conjugacy classes of maximal subgroups in a finite soluble group

Research Report MRR97-043

On the number of conjugacy classes of maximal subgroups in a finite soluble group

Burkhard Höfling

Abstract: We show that for many formations $\frak F$ , there exists an integer $n = \overline m(\frak F)$ such that every finite soluble group G not belonging to the class $\frak F$ has at most n conjugacy classes of maximal subgroups belonging to the class $\frak F$ . If $\frak F$ is a local formation with formation function f, we bound $\overline m ({\frak F})$ in terms of the $\overline m (f(p))$ $(p \in \Bbb P)$ . In particular, we show that $\overline m ({\frak N}^k) = k+1$ for every nonnegative integer k, where ${\frak N}^k$ is the class of all finite groups of Fitting length $\le k$ .

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