Partitions and permutations
The author is interested in the following problem: what information can we get on a permutation group of degree \(n\) knowing the set (or multiset) of partitions into cycles of its elements? Same question if we only know the set (or multiset) of partitions of \(n\), which arise as cycle partitions of its elements. The first constatation is that there exists non-isomorphic groups (even as abstract groups) with the same set of partitions. However two permutation groups with the same set of partitions have the same order, the same orbit partition and the same multiset of partitions. Hence the set and multiset of partitions carry the same information. The informations carried by the multiset of number partitions of \(G\) is the same as the information carried by the Pólya cycle index of \(G\), a polynomial in \(n\) indeterminates. Again there exists non-isomorphic groups (even as abstract groups) with the same cycle index. Two permutation groups with the same cycle index have the same number of orbits on \(k\)-tuples and on \(k\)-subsets, and the same order. Very little can be said when we know only the set of number partitions, without multiplicities. It determines the orders of elements of the group but not the order nor the number of orbits of the group. The author also considers the cycles of the permutations in isolation, this is linked to the Parker vector. The Parker vector is determined by the cycle index. It does not determine the order of the group. The author also investigates the same problem for infinite permutation groups. For instance he proves that if two infinite permutation groups have the same set of partitions and one of them is \(k\)-transitive (resp. highly transitive, oligomorphic, finitary, cofinitary), then so is the other one. The author gives several interesting open questions along the way, like this one: if \(H\) is a subgroup of \(G\) (possibly infinite) and they have the same set of partitions, are they equal?
scientific article; zbMATH DE number 2157537