Some relations for the lengths of orbits on k-sets and (k - 1)-sets

Abstract.

Let G be a permutation group on a finite set $\mit\Omega$ of size n. The paper is devoted to investigation of the relationship between the lengths of orbits on k-sets and (k - 1)-sets. The main results are the following:¶¶Theorem A. Let G be an Abelian or Hamiltonian group acting transitively on the finite set $\mit\Omega $ . Let $ \mit\Sigma \subseteq \mit\Omega $ be a k-set and $ \mit\Delta $ be a (k - 1)-subset of $\mit\Sigma $ . Then $ \mit\Sigma^G $ or $ \mit\Delta^G $ must be of the maximal length.¶¶Theorem B. Let G be a permutation group on the set $\mit\Omega $ and let $\mit\Sigma \subseteq \mit\Omega $ be a k-set, k > 2. Then there is a (k - 1)-set $\mit\Delta \subset \mit\Sigma $ such that¶¶$\left |\mit\Delta ^G\right |\ge {2 \over k^2 }\cdot {\left |\mit\Sigma ^G\right |}^{{k-1} \over {k}}$.¶¶Thus length is in some sense a hereditary property of orbits.