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A probabilistic approach to a classical result of ore
The subgroup commutativity degree sd(G) of a finite group G was introduced almost ten years ago and deals with the number of commuting subgroups in the subgroups lattice L(G) of G. The extremal case sd(G) = 1 detects a class of groups classified by Iwasawa in 1941 (in fact sd(G) represents a probabi...
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| Language: | English |
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Department of Mathematics and Applied Mathematics
2021
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| _version_ | 1867613226327343104 |
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| access_status_str | Open Access |
| author | Muhie, Seid Kassaw |
| author2 | Russo, Francesco G |
| author_browse | Muhie, Seid Kassaw Russo, Francesco G |
| author_facet | Russo, Francesco G Muhie, Seid Kassaw |
| author_sort | Muhie, Seid Kassaw |
| collection | Thesis |
| description | The subgroup commutativity degree sd(G) of a finite group G was introduced almost ten years ago and deals with the number of commuting subgroups in the subgroups lattice L(G) of G. The extremal case sd(G) = 1 detects a class of groups classified by Iwasawa in 1941 (in fact sd(G) represents a probabilistic measure which allows us to understand how far is G from the groups of Iwasawa). Among them we have sd(G) = 1 when L(G) is distributive, that is, when G is cyclic. The characterization of a cyclic group by the distributivity of its lattice of subgroups is due to a classical result of Ore in 1938. Therefore sd(G) is strongly related to structural properties of L(G). Here we introduce a new notion of probability gsd(G) in which two arbitrary sublattices S(G) and T(G) of L(G) are involved simultaneously. In case S(G) = T(G) = L(G), we find exactly sd(G). Upper and lower bounds in terms of gsd(G) and sd(G) are among our main contributions, when the condition S(G) = T(G) = L(G) is removed. Then we investigate the problem of counting the pairs of commuting subgroups via an appropriate graph. Looking at the literature, we noted that a similar problem motivated the permutability graph of non–normal subgroups ΓN (G) in 1995, that is, the graph where all proper non– normal subgroups of G form the vertex set of ΓN (G) and two vertices H and K are joined if HK = KH. The graph ΓN (G) has been recently generalized via the notion of permutability graph of subgroups Γ(G), extending the vertex set to all proper subgroups of G and keeping the same criterion to join two vertices. We use gsd(G), in order to introduce the non–permutability graph of subgroups ΓL(G) ; its vertices are now given by the set L(G) − CL(G)(L(G)), where CL(G)(L(G)) is the smallest sublattice of L(G) containing all permutable subgroups of G, and we join two vertices H, K of ΓL(G) if HK 6= KH. We finally study some classical invariants for ΓL(G) and find numerical relations between the number of edges of ΓL(G) and gsd(G). |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/33841 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:32:46.693Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2021 |
| publishDateRange | 2021 |
| publishDateSort | 2021 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/33841 A probabilistic approach to a classical result of ore Muhie, Seid Kassaw Russo, Francesco G Subgroup Lattices Subgroup commutativity degree Permutability graph Dihedral groups Polynomial functions The subgroup commutativity degree sd(G) of a finite group G was introduced almost ten years ago and deals with the number of commuting subgroups in the subgroups lattice L(G) of G. The extremal case sd(G) = 1 detects a class of groups classified by Iwasawa in 1941 (in fact sd(G) represents a probabilistic measure which allows us to understand how far is G from the groups of Iwasawa). Among them we have sd(G) = 1 when L(G) is distributive, that is, when G is cyclic. The characterization of a cyclic group by the distributivity of its lattice of subgroups is due to a classical result of Ore in 1938. Therefore sd(G) is strongly related to structural properties of L(G). Here we introduce a new notion of probability gsd(G) in which two arbitrary sublattices S(G) and T(G) of L(G) are involved simultaneously. In case S(G) = T(G) = L(G), we find exactly sd(G). Upper and lower bounds in terms of gsd(G) and sd(G) are among our main contributions, when the condition S(G) = T(G) = L(G) is removed. Then we investigate the problem of counting the pairs of commuting subgroups via an appropriate graph. Looking at the literature, we noted that a similar problem motivated the permutability graph of non–normal subgroups ΓN (G) in 1995, that is, the graph where all proper non– normal subgroups of G form the vertex set of ΓN (G) and two vertices H and K are joined if HK = KH. The graph ΓN (G) has been recently generalized via the notion of permutability graph of subgroups Γ(G), extending the vertex set to all proper subgroups of G and keeping the same criterion to join two vertices. We use gsd(G), in order to introduce the non–permutability graph of subgroups ΓL(G) ; its vertices are now given by the set L(G) − CL(G)(L(G)), where CL(G)(L(G)) is the smallest sublattice of L(G) containing all permutable subgroups of G, and we join two vertices H, K of ΓL(G) if HK 6= KH. We finally study some classical invariants for ΓL(G) and find numerical relations between the number of edges of ΓL(G) and gsd(G). 2021-08-31T08:33:30Z 2021-08-31T08:33:30Z 2021 2021-08-31T08:32:57Z Doctoral Thesis Doctoral PhD http://hdl.handle.net/11427/33841 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science |
| spellingShingle | Subgroup Lattices Subgroup commutativity degree Permutability graph Dihedral groups Polynomial functions Muhie, Seid Kassaw A probabilistic approach to a classical result of ore |
| thesis_degree_str | Doctoral |
| title | A probabilistic approach to a classical result of ore |
| title_full | A probabilistic approach to a classical result of ore |
| title_fullStr | A probabilistic approach to a classical result of ore |
| title_full_unstemmed | A probabilistic approach to a classical result of ore |
| title_short | A probabilistic approach to a classical result of ore |
| title_sort | probabilistic approach to a classical result of ore |
| topic | Subgroup Lattices Subgroup commutativity degree Permutability graph Dihedral groups Polynomial functions |
| url | http://hdl.handle.net/11427/33841 |
| work_keys_str_mv | AT muhieseidkassaw aprobabilisticapproachtoaclassicalresultofore AT muhieseidkassaw probabilisticapproachtoaclassicalresultofore |