## A probabilistic approach to a classical result of ore

dc.contributor.advisor | Russo, Francesco G | |

dc.contributor.author | Muhie, Seid Kassaw | |

dc.date.accessioned | 2021-08-31T08:33:30Z | |

dc.date.available | 2021-08-31T08:33:30Z | |

dc.date.issued | 2021 | |

dc.date.updated | 2021-08-31T08:32:57Z | |

dc.description.abstract | 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). | |

dc.identifier.apacitation | Muhie, S. K. (2021). <i>A probabilistic approach to a classical result of ore</i>. (). ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/33841 | en_ZA |

dc.identifier.chicagocitation | Muhie, Seid Kassaw. <i>"A probabilistic approach to a classical result of ore."</i> ., ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2021. http://hdl.handle.net/11427/33841 | en_ZA |

dc.identifier.citation | Muhie, S.K. 2021. A probabilistic approach to a classical result of ore. . ,Faculty of Science ,Department of Mathematics and Applied Mathematics. http://hdl.handle.net/11427/33841 | en_ZA |

dc.identifier.ris | TY - Doctoral Thesis AU - Muhie, Seid Kassaw AB - 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). DA - 2021 DB - OpenUCT DP - University of Cape Town KW - Subgroup Lattices KW - Subgroup commutativity degree KW - Permutability graph KW - Dihedral groups KW - Polynomial functions LK - https://open.uct.ac.za PY - 2021 T1 - A probabilistic approach to a classical result of ore TI - A probabilistic approach to a classical result of ore UR - http://hdl.handle.net/11427/33841 ER - | en_ZA |

dc.identifier.uri | http://hdl.handle.net/11427/33841 | |

dc.identifier.vancouvercitation | Muhie SK. A probabilistic approach to a classical result of ore. []. ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2021 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/33841 | en_ZA |

dc.language.rfc3066 | eng | |

dc.publisher.department | Department of Mathematics and Applied Mathematics | |

dc.publisher.faculty | Faculty of Science | |

dc.subject | Subgroup Lattices | |

dc.subject | Subgroup commutativity degree | |

dc.subject | Permutability graph | |

dc.subject | Dihedral groups | |

dc.subject | Polynomial functions | |

dc.title | A probabilistic approach to a classical result of ore | |

dc.type | Doctoral Thesis | |

dc.type.qualificationlevel | Doctoral | |

dc.type.qualificationlevel | PhD |