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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Normal subgroups of doubly transitive automorphism groups of chains


Authors: Richard N. Ball and Manfred Droste
Journal: Trans. Amer. Math. Soc. 290 (1985), 647-664
MSC: Primary 20B27; Secondary 06F15
DOI: https://doi.org/10.1090/S0002-9947-1985-0792817-5
MathSciNet review: 792817
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Abstract: We characterize the structure of the normal subgroup lattice of $ 2$-transitive automorphism groups $ A(\Omega )$ of infinite chains $ (\Omega , \leqslant )$ by the structure of the Dedekind completion $ (\bar \Omega , \leqslant )$ of the chain $ (\Omega , \leqslant )$. As a consequence we obtain various group-theoretical results on the normal subgroups of $ A(\Omega )$, including that any proper subnormal subgroup of $ A(\Omega )$ is indeed normal and contained in a maximal proper normal subgroup of $ A(\Omega )$, and that $ A(\Omega )$ has precisely $ 5$ normal subgroups if and only if the coterminality of the chain $ (\Omega , \leqslant )$ is countable.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0792817-5
Keywords: Order-preserving permutation, automorphism groups of chains, homogeneous linearly ordered sets, lattice-ordered groups, normal subgroup lattice, cofinality of a chain
Article copyright: © Copyright 1985 American Mathematical Society