# Subgroups of 3-Factor Direct Products

@article{Neuen2019SubgroupsO3,
title={Subgroups of 3-Factor Direct Products},
author={Daniel Neuen and Pascal Schweitzer},
journal={Tatra Mountains Mathematical Publications},
year={2019},
volume={73},
pages={19 - 38}
}
• Published 2019
• Mathematics, Biology
• Tatra Mountains Mathematical Publications
Abstract Extending Goursat’s Lemma we investigate the structure of subdirect products of 3-factor direct products. We construct several examples and then provide a structure theorem showing that every such group is essentially obtained by a combination of the examples. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat’s Lemma, for two special… Expand
3 Citations

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#### References

SHOWING 1-10 OF 15 REFERENCES
Structure and finiteness properties of subdirect products of groups
• Mathematics
• 2009
We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNNExpand
Subgroups of Finite Abelian Groups Having Rank Two via Goursat’s Lemma
Abstract Using Goursat’s lemma for groups, a simple representation and the invariant factor decompositions of the subgroups of the group Zm × Zn are deduced, where m and n are arbitrary positiveExpand
A generalized Goursat lemma
• Mathematics
• 2011
Abstract In this note the usual Goursat lemma, which describes subgroups of the direct product of two groups, is generalized to describing subgroups of a direct product A1 × A2 × · · · × An of aExpand
Counting Subgroups in a Direct Product of Finite Cyclic Groups
Summary We calculate the number of subgroups in a direct product of finite cyclic groups by applying the fundamental theorem of finite abelian groups and a well-known structure theorem due toExpand
The Theory Of Groups
Introduction Normal subgroups and homomorphisms Elementary theory of abelian groups Sylow theorems Permutation groups Automorphisms Free groups Lattices and composition series A theorem of FrobeniusExpand
On the finite presentation of subdirect products and the nature of residually free groups
We establish {\it virtual surjection to pairs} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if $\Gamma_1,\ldots,\Gamma_n$ are finitely presented andExpand
ON THE SUBGROUPS OF FINITE ABELIAN GROUPS OF RANK THREE
• Mathematics
• 2013
We describe the subgroups of the group Zm Zn Zr and derive a simple formula for the total number s(m;n;r) of the subgroups, where m;n;r are arbitrary positive integers. An asymptotic formula for theExpand
Subgroup Lattices and Symmetric Functions
Introduction Subgroups of finite Abelian groups Hall-Littlewood symmetric functions Some enumerative combinatorics Some algebraic combinatorics.
Subgroups of direct products of groups , ideals and subrings of direct products of rings , and Goursat ’ s lemma . In Rings , modules and representations , volume 480 of Contemp
• 2009
Subgroups of direct products of groups, ideals and subrings of direct products of rings, and Goursat’s lemma
• Rings, modules and representations, volume 480 of Contemp. Math., pages 1–12. Amer. Math. Soc., Providence, RI,
• 2009