Characteristic subgroups of latticeordered groups
Authors:
Richard D. Byrd, Paul Conrad and Justin T. Lloyd
Journal:
Trans. Amer. Math. Soc. 158 (1971), 339371
MSC:
Primary 06.75
MathSciNet review:
0279014
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Abstract: Characteristic subgroups of an group are those convex subgroups that are fixed by each automorphism. Certain sublattices of the lattice of all convex subgroups determine characteristic subgroups which we call socles. Various socles of an group are constructed and this construction leads to some structure theorems. The concept of a shifting subgroup is introduced and yields results relating the structure of an group to that of the lattice of characteristic subgroups. Interesting results are obtained when the group is characteristically simple. We investigate the characteristic subgroups of the vector lattice of realvalued functions on a root system and determine those vector lattices in which every ideal is characteristic. The automorphism group of the vector lattice of all continuous realvalued functions (almost finite realvalued functions) on a topological space (a Stone space) is shown to be a splitting extension of the polar preserving automorphisms by the ring automorphisms. This result allows us to construct characteristically simple vector lattices. We show that selfinjective vector lattices exist and that an archimedean selfinjective vector lattice is characteristically simple. It is proven that each group can be embedded as an subgroup of an algebraically simple group. In addition, we prove that each representable (abelian) group can be embedded as an subgroup of a characteristically simple representable (abelian) group.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197102790147
PII:
S 00029947(1971)02790147
Keywords:
Socles of an group,
automorphism,
characteristic subgroup,
characteristically simple group,
polar,
Boolean algebra,
completely reducible group,
shifting subgroup,
simple subgroup,
completely reducible group,
lexsubgroup,
prime subgroup,
closed subgroup,
cardinally indecomposable group,
lexextension,
basic element,
basis,
principal polar,
lexkernel,
regular subgroup,
special element,
finite valued group,
root system,
root,
vector lattice,
essential extension,
archimedean extension,
completely regular space,
real compact space,
splitting extension,
selfinjective group,
large subgroup,
hyperarchimedean group,
radical,
ideal radical,
distributive radical,
singular element
Article copyright:
© Copyright 1971
American Mathematical Society
