On the Wielandt length of a finite supersoluble group

Mathematics Research Report MRR96-029

On the Wielandt length of a finite supersoluble group

Asif Ali

Abstract: The main aim of this paper is to find a relation between the Wielandt length of a supersoluble group G and invariants of its Sylow subgroups. We are able to relate the Wielandt length of G with the Wielandt length of Sylow subgroups of G when the Wielandt length of Sylow subgroups is at most one or two. We prove that if all Sylow subgroups of G have Wielandt length at most n, then G has Wielandt length at most n+1 where n = 1,2. To find a similar result for supersoluble groups whose Sylow subgroups have higher Wielandt length seems difficult because more information is required about the Wielandt structure of nilpotent groups than is presently available. However we can bound the Wielandt length in terms of the nilpotency classes of the Sylow subgroups. We prove that if G is a supersoluble group and n is the maximum of the nilpotency classes of the Sylow subgroups of G, then $G \in \cal W$ $_{n+1}$ . We give examples which prove that this bound is best possible and that supersolubility of G is necessary in this result.

All these results are based on the calculation of the Wielandt subgroup and hence the Wielandt length of supersoluble groups. Then we use this information to prove that if a Sylow p-subgroup of a supersoluble group G has Wielandt length n >1, then a Sylow p-subgroup of $G/\omega (G)$ has Wielandt length at most n-1.

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