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Lattice embeddings of abelian prime power groups

Part of: Abelian groups Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

Roland Schmidt
Roland Schmidt
Affiliation:
Mathematisches Seminar Universität KielD-24098 Kiel, Germany
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Abstract

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We solve the following problem which was posed by Barnes in 1962. For which abelian groups G and H of the same prime power order is it possible to embed the subgroup lattice of G in that of H? It follows from Barnes' results and a theorem of Herrmann and Huhn that if there exists such an embedding and G contains three independent elements of order p2, then G and H are isomorphic. This reduces the problem to the case that G is the direct product of cyclic p-groups only two of which have order larger than p. We determine all groups H for which the desired embedding exists.

MSC classification

Secondary: 20D30: Series and lattices of subgroups 20K01: Finite abelian groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 2 , April 1997 , pp. 259 - 278
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Barnes, D. W., ‘Lattice embeddings of prime power groups’, J. Austral. Math. Soc. 2 (1962), 17–34.CrossRefGoogle Scholar
[2]Herrmann, C. and Huhn, P., ‘Zum Begriff der Charakteristik modularer Verbände’, Math. Z. 144 (1975), 185–194.CrossRefGoogle Scholar
[3]Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[4]Schmidt, R., Subgroup lattices of groups (De Gruyter, Berlin, 1994).CrossRefGoogle Scholar