Closed subgroups of latticeordered permutation groups
Author:
Stephen H. McCleary
Journal:
Trans. Amer. Math. Soc. 173 (1972), 303314
MSC:
Primary 06A55; Secondary 20B99
MathSciNet review:
0311535
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Abstract: Let be an subgroup of the latticeordered group of orderpreserving permutations of a chain ; and in this abstract, assume for convenience that is transitive. Let denote the completion by Dedekind cuts of . The stabilizer subgroups , will be used to characterize certain subgroups of which are closed (under arbitrary suprema which exist in ). If is an block of (a nonempty convex subset such that for any , either or is empty), and if will denote ; and the block system consisting of the translates of will be called closed if is closed. When the collection of block systems is totally ordered (by inclusion, viewing the systems as congruences), there is a smallest closed system , and all systems above are closed. is the trivial system (of singletons) iff is complete (in ). is closed iff is a cut in i.e., is not in the interior of any . Every closed convex subgroup of is an intersection of stabilizers of cuts in . Every closed prime subgroup is either a stabilizer of a cut in , or else is minimal and is the intersection of a tower of such stabilizers. is the distributive radical of , so that acts faithfully (and completely) on iff is completely distributive. Every closed ideal of is for some system . A group in which every nontrivial block supports some (e.g., a generalized ordered wreath product) fails to be complete iff has a smallest nontrivial system and the restriction is transitive and lacks elements of bounded support. These results about permutation groups are used to show that if is an abstract group having a representing subgroup, its closed ideals form a tower under inclusion; and that if is a Holland kernel of a completely distributive abstract group , then so is the set of closures , so that if has a transitive representation as a permutation group, it has a complete transitive representation.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197203115351
PII:
S 00029947(1972)03115351
Keywords:
Latticeordered permutation group,
totally ordered set,
complete subgroup,
prime subgroup,
closed subgroup,
stabilizer subgroup,
complete distributivity,
wreath product
Article copyright:
© Copyright 1972 American Mathematical Society
