Abstract
We give a category-free order theoretic variant of a key result in Auinger and Szendrei (J Pure Appl Algebra 204(3):493–506, 2006) and illustrate how it might be used to compute whether a finite X-generated group H admits a canonical dual prehomomorphism into the Margolis–Meakin expansion M(G) of a finite X-generated group G. We show that for G the Klein four-group a suitable H must be of exponent 6 at least and recapture a result of Szakács.
Highlights
The following note considers canonical, i.e. generator preserving dual prehomomorphisms from an X-generated group H into the Margolis–Meakin expansion M(G) of an X-generated group G
For G the Klein four-group, we prove that a suitable group H must be of exponent 6 at least and recapture a result of Szakács [6]
Note that a canonical dual prehomomorphism ∶ H → M(G) always induces a generator respecting homomorphism from H onto G, given by [w] ↦ w, which follows from the fact that in M(G) we have that (Γ(⟨v⟩), v) ≤ (Γ(⟨w⟩), w) implies v = w and respects generators
Summary
The following note considers canonical, i.e. generator preserving dual prehomomorphisms from an X-generated group H into the Margolis–Meakin expansion M(G) of an X-generated group G. We give a necessary and sufficient order theoretic condition for M(G) to admit a canonical dual prehomomorphism from an X-generated group H. Note that a canonical dual prehomomorphism ∶ H → M(G) always induces a generator respecting homomorphism from H onto G, given by [w] ↦ w , which follows from the fact that in M(G) we have that (Γ(⟨v⟩), v) ≤ (Γ(⟨w⟩), w) implies v = w and respects generators.
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