<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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 cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphisms between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups.</p></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 cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphisms between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups.

Subgroups of 3-Factor Direct Products

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Subgroups of 3-Factor Direct Products"}

Extending Goursat’s Lemma we investigate the structure of subdirect products of 3-factor direct products. We construct several examples and then provide a...

Subgroups of 3-Factor Direct Products

Published Online: Aug 15, 2019

Page range: 19 - 38

Received: Jul 25, 2018

DOI: https://doi.org/10.2478/tmmp-2019-0003

Keywords
subdirect products, Goursat’s Lemma, counting subgroups of products

© 2019 Daniel Neuen et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Subgroups of 3-Factor Direct Products

Published Online: Aug 15, 2019

Page range: 19 - 38

Received: Jul 25, 2018

DOI: https://doi.org/10.2478/tmmp-2019-0003

Keywordssubdirect products, Goursat’s Lemma, counting subgroups of products

© 2019 Daniel Neuen et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
subdirect products, Goursat’s Lemma, counting subgroups of products