Highest rank of a polytope for An

Peter J. Cameron; Maria Elisa Fernandes; Dimitri Leemans; Mark Mixer

doi:10.1112/plms.12039

Highest rank of a polytope for A_n

Peter J. Cameron, Maria Elisa Fernandes, Dimitri Leemans, Mark Mixer

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that the highest rank of a string C-group constructed from an alternating group A_n is 3 if n=5, 4 if n=9, 5 if n=10, 6 if n=11, and ⌊(n−1)/2⌋ if n⩾12. Moreover, if n=3,4,6,7, or 8, the group A_n is not a string C-group. This solves a conjecture made by the last three authors in 2012.

Original language	English
Pages (from-to)	135-176
Number of pages	42
Journal	Proceedings of the London Mathematical Society
Volume	115
Issue number	1
Early online date	28 Apr 2017
DOIs	https://doi.org/10.1112/plms.12039
Publication status	Published - 4 Jul 2017

Keywords

Abstract regular polytypes
String C-groups
Alternating groups
Permutation groups

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10.1112/plms.12039

Cameron_2017_Highest_PLMS_AAM
© 2017, London Mathematical Society. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1112/plms.12039
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Cite this

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title = "Highest rank of a polytope for An",

abstract = "We prove that the highest rank of a string C-group constructed from an alternating group An is 3 if n=5, 4 if n=9, 5 if n=10, 6 if n=11, and ⌊(n−1)/2⌋ if n⩾12. Moreover, if n=3,4,6,7, or 8, the group An is not a string C-group. This solves a conjecture made by the last three authors in 2012.",

keywords = "Abstract regular polytypes, String C-groups, Alternating groups, Permutation groups",

author = "Cameron, \{Peter J.\} and Fernandes, \{Maria Elisa\} and Dimitri Leemans and Mark Mixer",

note = "This research was supported by a Marsden grant (UOA1218) of the Royal Society of New Zealand, and by the Portuguese Foundation for Science and Technology (FCT-Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.",

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AU - Cameron, Peter J.

AU - Fernandes, Maria Elisa

AU - Leemans, Dimitri

AU - Mixer, Mark

N1 - This research was supported by a Marsden grant (UOA1218) of the Royal Society of New Zealand, and by the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013.

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Y1 - 2017/7/4

N2 - We prove that the highest rank of a string C-group constructed from an alternating group An is 3 if n=5, 4 if n=9, 5 if n=10, 6 if n=11, and ⌊(n−1)/2⌋ if n⩾12. Moreover, if n=3,4,6,7, or 8, the group An is not a string C-group. This solves a conjecture made by the last three authors in 2012.

AB - We prove that the highest rank of a string C-group constructed from an alternating group An is 3 if n=5, 4 if n=9, 5 if n=10, 6 if n=11, and ⌊(n−1)/2⌋ if n⩾12. Moreover, if n=3,4,6,7, or 8, the group An is not a string C-group. This solves a conjecture made by the last three authors in 2012.

KW - Abstract regular polytypes

KW - String C-groups

KW - Alternating groups

KW - Permutation groups

UR - https://www.scopus.com/pages/publications/85021787703

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DO - 10.1112/plms.12039

M3 - Article

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SP - 135

EP - 176

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 1

ER -

Highest rank of a polytope for A_n

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Highest rank of a polytope for A_n