On the Wielandt length of subgroups of finite soluble groups

Mathematics Research Report MRR96-040

On the Wielandt length of subgroups of finite soluble groups

Asif Ali

Abstract: In this paper we investigate the class $\cal K$ of soluble groups G such that each subgroup H of G has Wielandt length at most the Wielandt length of G. We conjecture that a soluble group G belongs to $\cal K$ if and only if all Sylow subgroups of G have Wielandt length at most the Wielandt length of G. (We denote by $\cal V$ the class of groups G whose all Sylow subgroups have Wielandt length at most the Wielandt length of G). In other words we conjecture that $\cal V = \cal K$ . It is easy to see that $\cal K \subseteq \cal V$ but the converse seems hard to prove. We prove in this paper that if $\cal L$ is the class of soluble groups G which satisfy the condition that for each subgroup H and a normal subgroup N of G, we have $HN/N \cap \omega (G/N) \subseteq \omega (HN/N)$ , then $\cal L \subseteq \cal K$ . We know that this condition is not satisfied by all groups $G \in \cal K$ (for example

). However we are able to prove that if $\cal R$ denotes the class of soluble groups having p-length one for all primes p then $\cal L = \cal R$ .

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