arXiv subjects
Publications and source records attributed to Csaba Schneider.
We study Cartesian decompositions of sets that are acted upon intransitively by innately transitive permutation groups. We prove that such groups have at most three orbits on such a decomposition. A consequence of this result is that if $G$ is an innately transitive subgroup of a wreath product in product action then the natural projection of $G$ into the top group has at most 2 orbits.
A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the plinth is simple. Here we extend that classification and identify several different types of Cartesian decompositions that can be preserved by an innately transitive group with a non-abelian plinth. These different types of decompositions lead to different types of embeddings of the acting group into wreath products in product action. We also obtain a full characterisation of embeddings of innately transitive groups with diagonal type into such wreath products.
Our aim is to describe the theory of Cartesian decompositions preserved by some member of a large family of finite transitive permutation groups called innatelytransitive groups.
A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a definition and detailed examination of `Cartesian decompositions' of the permuted set, relating them to certain `Cartesian systemsof subgroups'. These concepts, and the bijective connections between them, are explored in greater generality, with specific future applications in mind.
For odd primes we prove some structure theorems for finite $p$-groups $G$, such that $G''\neq 1$ and $|G'/G''|=p^3$. Building on results of Blackburn and Hall, it is shown that $\lcs G3$ is a maximal subgroup of $G'$, the group $G$ has a central decomposition into two simpler subgroups, and, moreover, $G'$ has one of two isomorphism types.
A nilpotent quotient algorithm for finitely presented Lie rings over Z (LieNQ) is described. The paper studies graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. The nilpotent presentation consists of generators for the abelian group and the products---expressed as linear combinations---for pairs formed by generators. Using that presentation the word problem is decidable in $L$. Provided that the Lie ring $L$ is graded, it is possible to determine the canonical presentation for a lower central factor of $L$. LieNQ's complexity is studied and it is shown that optimizing the presentation is NP-hard. Computational details are provided with examples, timing and some structure theorems obtained from computations. Implementation in C and GAP 3.5 interface is available.