Abstract
Let G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice mathcal {L}(H) embeds in mathcal {L}(G). It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.
Highlights
Mathematics Subject Classification 20E15 · 06B15. The aim of this short note is to define a sublattice embedding property which, into the universe of subgroup lattices of abelian groups, is satisfied precisely by all the elements corresponding to pure subgroups
Recall that a subgroup H of an abelian group G is said pure in G if Gn ∩ H = H n for all positive integers n, or, equivalently, if every h ∈ H having a nth root in G admits a nth root in H
Trombetti the recent survey [2]), where he describes the structure of abelian groups whose pure subgroups admit a complement
Summary
The aim of this short note is to define a sublattice embedding property which, into the universe of subgroup lattices of abelian groups, is satisfied precisely by all the elements corresponding to pure subgroups. We say that c and d have the same cyclic structure if there exists a lattice isomorphism φ : [c/0] −→ [d/0] such that x does not properly split in x ∨ φ(x) whenever [x/0] is a chain.
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