Independence complexes of finite groups
| dc.contributor.author | Pinckney, Casey M., author | |
| dc.contributor.author | Hulpke, Alexander, advisor | |
| dc.contributor.author | Peterson, Chris, advisor | |
| dc.contributor.author | Adams, Henry, committee member | |
| dc.contributor.author | Neilson, James, committee member | |
| dc.date.accessioned | 2022-01-07T11:30:06Z | |
| dc.date.available | 2022-01-07T11:30:06Z | |
| dc.date.issued | 2021 | |
| dc.description.abstract | Understanding generating sets for finite groups has been explored previously via the generating graph of a group, where vertices are group elements and edges are given by pairs of group elements that generate the group. We generalize this idea by considering minimal generating sets (with respect to inclusion) for subgroups of finite groups. These form a simplicial complex, which we call the independence complex. The vertices of the independence complex are nonidentity group elements and the faces of size k correspond to minimal generating sets of size k. We give a complete characterization via constructive algorithms, together with enumeration results, for the independence complexes of cyclic groups whose order is a squarefree product of primes, finite abelian groups whose order is a product of powers of distinct primes, and the nonabelian class of semidirect products Cp1p3…p2n-1 rtimes Cp2p4…p2n where p1,p2,…,p2n are distinct primes with p2i-1 > p2i for all 1 ≤ i ≤ n. In the latter case, we introduce a tool called a combinatorial diagram, which is a multipartite simplicial complex under certain numerical and minimal covering conditions. Combinatorial diagrams seem to be an interesting area of study on their own. We also include GAP and Polymake code which generates the facets of any (small enough) finite group, as well as visualize the independence complexes in small dimensions. | |
| dc.format.medium | born digital | |
| dc.format.medium | doctoral dissertations | |
| dc.identifier | Pinckney_colostate_0053A_16824.pdf | |
| dc.identifier.uri | https://hdl.handle.net/10217/234240 | |
| dc.language | English | |
| dc.language.iso | eng | |
| dc.publisher | Colorado State University. Libraries | |
| dc.relation.ispartof | 2020- | |
| dc.rights | Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. | |
| dc.subject | finite groups | |
| dc.subject | independent sets | |
| dc.subject | simplicial complexes | |
| dc.subject | independence complexes | |
| dc.subject | clutters | |
| dc.subject | minimal generating sets | |
| dc.title | Independence complexes of finite groups | |
| dc.type | Text | |
| dcterms.rights.dpla | This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Colorado State University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy (Ph.D.) |
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