On groups all of whose Haar graphs are Cayley graphs
Abstract A Cayley graph of a group H is a finite simple graph $$\Gamma $$ such that $$\mathrm{Aut}(\Gamma )$$ contains a subgroup isomorphic to H acting regularly on $$V(\Gamma ),$$ while a Haar graph of H is a finite simple bipartite graph $$\Sigma $$ such that $$\mathrm{Aut}(\Sigma )$$ contains a...
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Gespeichert in:
| Autor*in: |
Feng, Yan-Quan [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2019 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media, LLC, part of Springer Nature 2019 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of algebraic combinatorics - Springer US, 1992, 52(2019), 1 vom: 10. Juni, Seite 59-76 |
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| Übergeordnetes Werk: |
volume:52 ; year:2019 ; number:1 ; day:10 ; month:06 ; pages:59-76 |
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| DOI / URN: |
10.1007/s10801-019-00894-7 |
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| 520 | |a Abstract A Cayley graph of a group H is a finite simple graph $$\Gamma $$ such that $$\mathrm{Aut}(\Gamma )$$ contains a subgroup isomorphic to H acting regularly on $$V(\Gamma ),$$ while a Haar graph of H is a finite simple bipartite graph $$\Sigma $$ such that $$\mathrm{Aut}(\Sigma )$$ contains a subgroup isomorphic to H acting semiregularly on $$V(\Sigma )$$ and the H-orbits are equal to the bipartite sets of $$\Sigma $$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that $$D_6, \, D_8, \, D_{10}$$ and $$Q_8$$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs. (A group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph. | ||
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| 700 | 1 | |a Yang, Da-Wei |4 aut | |
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10.1007/s10801-019-00894-7 doi (DE-627)OLC2077611650 (DE-He213)s10801-019-00894-7-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Feng, Yan-Quan verfasserin aut On groups all of whose Haar graphs are Cayley graphs 2019 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract A Cayley graph of a group H is a finite simple graph $$\Gamma $$ such that $$\mathrm{Aut}(\Gamma )$$ contains a subgroup isomorphic to H acting regularly on $$V(\Gamma ),$$ while a Haar graph of H is a finite simple bipartite graph $$\Sigma $$ such that $$\mathrm{Aut}(\Sigma )$$ contains a subgroup isomorphic to H acting semiregularly on $$V(\Sigma )$$ and the H-orbits are equal to the bipartite sets of $$\Sigma $$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that $$D_6, \, D_8, \, D_{10}$$ and $$Q_8$$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs. (A group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph. Haar graph Cayley graph Vertex-transitive graph Kovács, István aut Yang, Da-Wei aut Enthalten in Journal of algebraic combinatorics Springer US, 1992 52(2019), 1 vom: 10. Juni, Seite 59-76 (DE-627)131180770 (DE-600)1143271-8 (DE-576)033043701 0925-9899 nnns volume:52 year:2019 number:1 day:10 month:06 pages:59-76 https://doi.org/10.1007/s10801-019-00894-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 52 2019 1 10 06 59-76 |
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10.1007/s10801-019-00894-7 doi (DE-627)OLC2077611650 (DE-He213)s10801-019-00894-7-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Feng, Yan-Quan verfasserin aut On groups all of whose Haar graphs are Cayley graphs 2019 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract A Cayley graph of a group H is a finite simple graph $$\Gamma $$ such that $$\mathrm{Aut}(\Gamma )$$ contains a subgroup isomorphic to H acting regularly on $$V(\Gamma ),$$ while a Haar graph of H is a finite simple bipartite graph $$\Sigma $$ such that $$\mathrm{Aut}(\Sigma )$$ contains a subgroup isomorphic to H acting semiregularly on $$V(\Sigma )$$ and the H-orbits are equal to the bipartite sets of $$\Sigma $$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that $$D_6, \, D_8, \, D_{10}$$ and $$Q_8$$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs. (A group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph. Haar graph Cayley graph Vertex-transitive graph Kovács, István aut Yang, Da-Wei aut Enthalten in Journal of algebraic combinatorics Springer US, 1992 52(2019), 1 vom: 10. Juni, Seite 59-76 (DE-627)131180770 (DE-600)1143271-8 (DE-576)033043701 0925-9899 nnns volume:52 year:2019 number:1 day:10 month:06 pages:59-76 https://doi.org/10.1007/s10801-019-00894-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 52 2019 1 10 06 59-76 |
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10.1007/s10801-019-00894-7 doi (DE-627)OLC2077611650 (DE-He213)s10801-019-00894-7-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Feng, Yan-Quan verfasserin aut On groups all of whose Haar graphs are Cayley graphs 2019 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract A Cayley graph of a group H is a finite simple graph $$\Gamma $$ such that $$\mathrm{Aut}(\Gamma )$$ contains a subgroup isomorphic to H acting regularly on $$V(\Gamma ),$$ while a Haar graph of H is a finite simple bipartite graph $$\Sigma $$ such that $$\mathrm{Aut}(\Sigma )$$ contains a subgroup isomorphic to H acting semiregularly on $$V(\Sigma )$$ and the H-orbits are equal to the bipartite sets of $$\Sigma $$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that $$D_6, \, D_8, \, D_{10}$$ and $$Q_8$$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs. (A group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph. Haar graph Cayley graph Vertex-transitive graph Kovács, István aut Yang, Da-Wei aut Enthalten in Journal of algebraic combinatorics Springer US, 1992 52(2019), 1 vom: 10. Juni, Seite 59-76 (DE-627)131180770 (DE-600)1143271-8 (DE-576)033043701 0925-9899 nnns volume:52 year:2019 number:1 day:10 month:06 pages:59-76 https://doi.org/10.1007/s10801-019-00894-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 52 2019 1 10 06 59-76 |
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10.1007/s10801-019-00894-7 doi (DE-627)OLC2077611650 (DE-He213)s10801-019-00894-7-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Feng, Yan-Quan verfasserin aut On groups all of whose Haar graphs are Cayley graphs 2019 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract A Cayley graph of a group H is a finite simple graph $$\Gamma $$ such that $$\mathrm{Aut}(\Gamma )$$ contains a subgroup isomorphic to H acting regularly on $$V(\Gamma ),$$ while a Haar graph of H is a finite simple bipartite graph $$\Sigma $$ such that $$\mathrm{Aut}(\Sigma )$$ contains a subgroup isomorphic to H acting semiregularly on $$V(\Sigma )$$ and the H-orbits are equal to the bipartite sets of $$\Sigma $$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that $$D_6, \, D_8, \, D_{10}$$ and $$Q_8$$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs. (A group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph. Haar graph Cayley graph Vertex-transitive graph Kovács, István aut Yang, Da-Wei aut Enthalten in Journal of algebraic combinatorics Springer US, 1992 52(2019), 1 vom: 10. Juni, Seite 59-76 (DE-627)131180770 (DE-600)1143271-8 (DE-576)033043701 0925-9899 nnns volume:52 year:2019 number:1 day:10 month:06 pages:59-76 https://doi.org/10.1007/s10801-019-00894-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 AR 52 2019 1 10 06 59-76 |
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Abstract A Cayley graph of a group H is a finite simple graph $$\Gamma $$ such that $$\mathrm{Aut}(\Gamma )$$ contains a subgroup isomorphic to H acting regularly on $$V(\Gamma ),$$ while a Haar graph of H is a finite simple bipartite graph $$\Sigma $$ such that $$\mathrm{Aut}(\Sigma )$$ contains a subgroup isomorphic to H acting semiregularly on $$V(\Sigma )$$ and the H-orbits are equal to the bipartite sets of $$\Sigma $$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that $$D_6, \, D_8, \, D_{10}$$ and $$Q_8$$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs. (A group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph. © Springer Science+Business Media, LLC, part of Springer Nature 2019 |
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Abstract A Cayley graph of a group H is a finite simple graph $$\Gamma $$ such that $$\mathrm{Aut}(\Gamma )$$ contains a subgroup isomorphic to H acting regularly on $$V(\Gamma ),$$ while a Haar graph of H is a finite simple bipartite graph $$\Sigma $$ such that $$\mathrm{Aut}(\Sigma )$$ contains a subgroup isomorphic to H acting semiregularly on $$V(\Sigma )$$ and the H-orbits are equal to the bipartite sets of $$\Sigma $$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that $$D_6, \, D_8, \, D_{10}$$ and $$Q_8$$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs. (A group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph. © Springer Science+Business Media, LLC, part of Springer Nature 2019 |
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Abstract A Cayley graph of a group H is a finite simple graph $$\Gamma $$ such that $$\mathrm{Aut}(\Gamma )$$ contains a subgroup isomorphic to H acting regularly on $$V(\Gamma ),$$ while a Haar graph of H is a finite simple bipartite graph $$\Sigma $$ such that $$\mathrm{Aut}(\Sigma )$$ contains a subgroup isomorphic to H acting semiregularly on $$V(\Sigma )$$ and the H-orbits are equal to the bipartite sets of $$\Sigma $$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that $$D_6, \, D_8, \, D_{10}$$ and $$Q_8$$ are the only finite inner abelian groups all of whose Haar graphs are Cayley graphs. (A group is called inner abelian if it is non-abelian, but all of its proper subgroups are abelian.) This result will then be used to derive that every non-solvable group admits a non-Cayley Haar graph. © Springer Science+Business Media, LLC, part of Springer Nature 2019 |
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On groups all of whose Haar graphs are Cayley graphs |
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Kovács, István Yang, Da-Wei |
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