Normal edge-transitive Cayley graphs and Frattini-like subgroups
For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generato...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Khosravi, Behnam [verfasserIn] |
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E-Artikel |
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| Sprache: |
Englisch |
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2022transfer abstract |
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| Umfang: |
26 |
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| Übergeordnetes Werk: |
Enthalten in: A two stage successive estimation based maximum power point tracking technique for photovoltaic modules - Christy Mano Raj, J.S. ELSEVIER, 2014, San Diego, Calif |
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| Übergeordnetes Werk: |
volume:607 ; year:2022 ; day:1 ; month:10 ; pages:473-498 ; extent:26 |
| Links: |
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| DOI / URN: |
10.1016/j.jalgebra.2021.03.035 |
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ELV058177965 |
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| 520 | |a For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. | ||
| 520 | |a For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. | ||
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10.1016/j.jalgebra.2021.03.035 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001806.pica (DE-627)ELV058177965 (ELSEVIER)S0021-8693(21)00207-6 DE-627 ger DE-627 rakwb eng 530 VZ 333.7 610 VZ 44.13 bkl Khosravi, Behnam verfasserin aut Normal edge-transitive Cayley graphs and Frattini-like subgroups 2022transfer abstract 26 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. secondary Elsevier 08A35 Elsevier 08A30 Elsevier primary Elsevier Praeger, Cheryl E. oth Enthalten in Elsevier Christy Mano Raj, J.S. ELSEVIER A two stage successive estimation based maximum power point tracking technique for photovoltaic modules 2014 San Diego, Calif (DE-627)ELV018117414 volume:607 year:2022 day:1 month:10 pages:473-498 extent:26 https://doi.org/10.1016/j.jalgebra.2021.03.035 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U GBV_ILN_40 GBV_ILN_2532 44.13 Medizinische Ökologie VZ AR 607 2022 1 1001 473-498 26 |
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10.1016/j.jalgebra.2021.03.035 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001806.pica (DE-627)ELV058177965 (ELSEVIER)S0021-8693(21)00207-6 DE-627 ger DE-627 rakwb eng 530 VZ 333.7 610 VZ 44.13 bkl Khosravi, Behnam verfasserin aut Normal edge-transitive Cayley graphs and Frattini-like subgroups 2022transfer abstract 26 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. secondary Elsevier 08A35 Elsevier 08A30 Elsevier primary Elsevier Praeger, Cheryl E. oth Enthalten in Elsevier Christy Mano Raj, J.S. ELSEVIER A two stage successive estimation based maximum power point tracking technique for photovoltaic modules 2014 San Diego, Calif (DE-627)ELV018117414 volume:607 year:2022 day:1 month:10 pages:473-498 extent:26 https://doi.org/10.1016/j.jalgebra.2021.03.035 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U GBV_ILN_40 GBV_ILN_2532 44.13 Medizinische Ökologie VZ AR 607 2022 1 1001 473-498 26 |
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10.1016/j.jalgebra.2021.03.035 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001806.pica (DE-627)ELV058177965 (ELSEVIER)S0021-8693(21)00207-6 DE-627 ger DE-627 rakwb eng 530 VZ 333.7 610 VZ 44.13 bkl Khosravi, Behnam verfasserin aut Normal edge-transitive Cayley graphs and Frattini-like subgroups 2022transfer abstract 26 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. secondary Elsevier 08A35 Elsevier 08A30 Elsevier primary Elsevier Praeger, Cheryl E. oth Enthalten in Elsevier Christy Mano Raj, J.S. ELSEVIER A two stage successive estimation based maximum power point tracking technique for photovoltaic modules 2014 San Diego, Calif (DE-627)ELV018117414 volume:607 year:2022 day:1 month:10 pages:473-498 extent:26 https://doi.org/10.1016/j.jalgebra.2021.03.035 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U GBV_ILN_40 GBV_ILN_2532 44.13 Medizinische Ökologie VZ AR 607 2022 1 1001 473-498 26 |
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10.1016/j.jalgebra.2021.03.035 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001806.pica (DE-627)ELV058177965 (ELSEVIER)S0021-8693(21)00207-6 DE-627 ger DE-627 rakwb eng 530 VZ 333.7 610 VZ 44.13 bkl Khosravi, Behnam verfasserin aut Normal edge-transitive Cayley graphs and Frattini-like subgroups 2022transfer abstract 26 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. secondary Elsevier 08A35 Elsevier 08A30 Elsevier primary Elsevier Praeger, Cheryl E. oth Enthalten in Elsevier Christy Mano Raj, J.S. ELSEVIER A two stage successive estimation based maximum power point tracking technique for photovoltaic modules 2014 San Diego, Calif (DE-627)ELV018117414 volume:607 year:2022 day:1 month:10 pages:473-498 extent:26 https://doi.org/10.1016/j.jalgebra.2021.03.035 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U GBV_ILN_40 GBV_ILN_2532 44.13 Medizinische Ökologie VZ AR 607 2022 1 1001 473-498 26 |
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10.1016/j.jalgebra.2021.03.035 doi /cbs_pica/cbs_olc/import_discovery/elsevier/einzuspielen/GBV00000000001806.pica (DE-627)ELV058177965 (ELSEVIER)S0021-8693(21)00207-6 DE-627 ger DE-627 rakwb eng 530 VZ 333.7 610 VZ 44.13 bkl Khosravi, Behnam verfasserin aut Normal edge-transitive Cayley graphs and Frattini-like subgroups 2022transfer abstract 26 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. secondary Elsevier 08A35 Elsevier 08A30 Elsevier primary Elsevier Praeger, Cheryl E. oth Enthalten in Elsevier Christy Mano Raj, J.S. ELSEVIER A two stage successive estimation based maximum power point tracking technique for photovoltaic modules 2014 San Diego, Calif (DE-627)ELV018117414 volume:607 year:2022 day:1 month:10 pages:473-498 extent:26 https://doi.org/10.1016/j.jalgebra.2021.03.035 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U GBV_ILN_40 GBV_ILN_2532 44.13 Medizinische Ökologie VZ AR 607 2022 1 1001 473-498 26 |
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A two stage successive estimation based maximum power point tracking technique for photovoltaic modules |
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Normal edge-transitive Cayley graphs and Frattini-like subgroups |
| abstract |
For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. |
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For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. |
| abstract_unstemmed |
For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transitive Cayley graphs of characteristically simple groups as building blocks for normal edge-transitive Cayley graphs whenever we have Φ ( G ; C ) trivial. We explore several questions which these results raise, some concerned with the set of all inverse-closed generating sets for groups in a given family. In particular we use this theory to classify all 4-valent normal edge-transitive Cayley graphs for dihedral groups; this involves a new construction of an infinite family of examples, and disproves a conjecture of Talebi. |
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