Abstract
When a group of automorphisms Δ acts on a module M, each invariant submodule H of M determines two normal subgroups of Δ: the centraliser of H and the centraliser of M/H. Similarly, each normal subgroup Г of Δ determines two invariant submodules of M: the fixed module of Г and the residual of Г. These connections determine Galois correspondences between the lattice of normal subgroups of Δ and lattice of invariant submodules of M.
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References
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, (1973).
G. Birkhoff, Lattice Theory, American Mathematical Society Colloquium Publications, Vol. 25 (1967).
L. Fuchs, Infinite Abelian Groups, Vols. I and II, Academic Press (1970–1973).
L. Fuchs, On Subdirect Unions, Acta Math. Acad. Sci. Hung. 3 (1952), 103–119.
J. Hausen and P. Schultz, The maximal normal p-subgroup of the automorphism group of an abelian p-group, Proc. Amer. Math. Soc 216 (1998), 2525–2533.
B. Huppert and N. Blackburn, Finite Groups, Vols. I, II and III, Springer-Verlag (1962–1973).
A. Mader and P. Schultz, Endomorphism rings and automorphism groups of almost completely decomposable groups, Comm. in Algebra, (to appear).
P. Schultz, When is an abelian p-group determined by the Jacobson radical of its endomorphism ring?, Abelian Group Theory and Related Topics: Conference on Abelian Groups August 1–7, 1993 Oberwolfach, Contemporary Mathematics Vol. 171 (1994), 385–396.
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Schultz, P. (1999). Groups acting on modules. In: Eklof, P.C., Göbel, R. (eds) Abelian Groups and Modules. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7591-2_6
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DOI: https://doi.org/10.1007/978-3-0348-7591-2_6
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7593-6
Online ISBN: 978-3-0348-7591-2
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