On orbit equivalent, two-step imprimitive permutation groups. (English) Zbl 1171.20003
Kappe, Luise-Charlotte (ed.) et al., Computational group theory and the theory of groups. AMS special session on computational group theory, Davidson, NC, USA, March 3–4, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4365-9/pbk). Contemporary Mathematics 470, 271-286 (2008).
Two permutation groups in \(\text{Sym}(\Omega)\) are called ‘orbit equivalent’ if they have the same orbits on the power set \(P(\Omega)\). It is easily seen that if \(G\) and \(H\) are orbit equivalent, then \(G\) is primitive if and only if so is \(H\). Pairs of primitive orbit equivalent groups were completely determined by Á. Seress, [Bull. Lond. Math. Soc. 29, No. 6, 697-704 (1997; Zbl 0892.20002)].
A transitive permutation group \(G\) is called ‘two-step imprimitive’ if there exists a nontrivial block system such that the actions of \(G\) on the set of blocks and of a block stabiliser on that block are both primitive. If \(G\) and \(H\) are orbit equivalent, then any system of imprimitivity for \(G\) is also a system of imprimitivity for \(H\). The authors also show that the actions of \(G\) and \(H\) on that block system are orbit equivalent, and the actions of the stabilisers \(G_B\) and \(H_B\) of a block \(B\) on \(B\) are also orbit equivalent. Therefore if \(G\) and \(H\) are orbit equivalent, then \(G\) is two-step imprimitive if and only if so is \(H\).
This paper starts the investigation of orbit equivalent pairs of two-step imprimitive groups. It is actually enough to determine the pairs such that one is a subgroup of the other. The study uses the fact that if \(G\) has a regular orbit in the power set then no proper subgroup of \(G\) is orbit equivalent to \(G\).
For the entire collection see [Zbl 1147.20002].
A transitive permutation group \(G\) is called ‘two-step imprimitive’ if there exists a nontrivial block system such that the actions of \(G\) on the set of blocks and of a block stabiliser on that block are both primitive. If \(G\) and \(H\) are orbit equivalent, then any system of imprimitivity for \(G\) is also a system of imprimitivity for \(H\). The authors also show that the actions of \(G\) and \(H\) on that block system are orbit equivalent, and the actions of the stabilisers \(G_B\) and \(H_B\) of a block \(B\) on \(B\) are also orbit equivalent. Therefore if \(G\) and \(H\) are orbit equivalent, then \(G\) is two-step imprimitive if and only if so is \(H\).
This paper starts the investigation of orbit equivalent pairs of two-step imprimitive groups. It is actually enough to determine the pairs such that one is a subgroup of the other. The study uses the fact that if \(G\) has a regular orbit in the power set then no proper subgroup of \(G\) is orbit equivalent to \(G\).
For the entire collection see [Zbl 1147.20002].
Reviewer: Alice Devillers (Western Australia)
