Abstract
If X is a set, the fix-set quasiorder on a group of permutations of X is the quasiorder induced by containment of the fix-sets of elements of S X . Axioms for such quasiorders on groups have previously been given. We generalise these to allow non-faithful group actions, the resulting abstract quasiorders being called fix-orders. We characterise the possible fix-orders on a given group G in terms of certain families of subgroups of G. The special case in which the members of the defining family of subgroups are all normal is considered. Software is used to construct and analyse the lattices of fix-orders of many small finite groups.
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References
Fearnley-Sander D., Stokes T.: Equality algebras. Bull. Austral. Math. Soc. 56, 177–191 (1997)
The GAP Group: GAP – Groups, Algorithms, and Programming, version 4.4.12 (2008). http://www.gap-system.org
Stokes T.: Axioms for function semigroups with agreement quasi-order. Algebra Universalis 66, 85–98 (2011)
Stokes T.: Rings with kernel inclusion quasi-order. Algebra Universalis 70, 379–391 (2013)
Schein B.M.: Relation algebras and function semigroups. Semigroup Forum 1, 1–62 (1970)
Schein B.M.: Lectures on semigroups of transformations. Amer. Math. Soc. Transl. ser. 2 113, 123–181 (1979)
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Presented by G. McNulty.
Dedicated to Brian Davey on the occasion of his 65th birthday
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Hawthorn, I., Manoharan, S. & Stokes, T. Groups with fix-set quasi-order. Algebra Univers. 74, 229–239 (2015). https://doi.org/10.1007/s00012-015-0349-9
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DOI: https://doi.org/10.1007/s00012-015-0349-9