Strong digraph groups
A digraph group is a group defined by non-empty presentation with the property that each relator is of the form $R(x, y)$ , where x and y are distinct generators and $R(\cdot , \cdot )$ is determined by some fixed cyclically reduced word $R(a, b)$ that involves both a and b. Associated with each suc...
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| Veröffentlicht in: | Canadian mathematical bulletin Jg. 67; H. 4; S. 991 - 1000 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Canada
Canadian Mathematical Society
01.12.2024
Cambridge University Press |
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| ISSN: | 0008-4395, 1496-4287 |
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| Abstract | A digraph group is a group defined by non-empty presentation with the property that each relator is of the form
$R(x, y)$
, where x and y are distinct generators and
$R(\cdot , \cdot )$
is determined by some fixed cyclically reduced word
$R(a, b)$
that involves both a and b. Associated with each such presentation is a digraph whose vertices correspond to the generators and whose arcs correspond to the relators. In this article, we consider digraph groups for strong digraphs that are digon-free and triangle-free. We classify when the digraph group is finite and show that in these cases it is cyclic, giving its order. We apply this result to the Cayley digraph of the generalized quaternion group, to circulant digraphs, and to Cartesian and direct products of strong digraphs. |
|---|---|
| AbstractList | A digraph group is a group defined by non-empty presentation with the property that each relator is of the form
$R(x, y)$
, where
x
and
y
are distinct generators and
$R(\cdot , \cdot )$
is determined by some fixed cyclically reduced word
$R(a, b)$
that involves both
a
and
b
. Associated with each such presentation is a digraph whose vertices correspond to the generators and whose arcs correspond to the relators. In this article, we consider digraph groups for strong digraphs that are digon-free and triangle-free. We classify when the digraph group is finite and show that in these cases it is cyclic, giving its order. We apply this result to the Cayley digraph of the generalized quaternion group, to circulant digraphs, and to Cartesian and direct products of strong digraphs. A digraph group is a group defined by non-empty presentation with the property that each relator is of the form $R(x, y)$ , where x and y are distinct generators and $R(\cdot , \cdot )$ is determined by some fixed cyclically reduced word $R(a, b)$ that involves both a and b. Associated with each such presentation is a digraph whose vertices correspond to the generators and whose arcs correspond to the relators. In this article, we consider digraph groups for strong digraphs that are digon-free and triangle-free. We classify when the digraph group is finite and show that in these cases it is cyclic, giving its order. We apply this result to the Cayley digraph of the generalized quaternion group, to circulant digraphs, and to Cartesian and direct products of strong digraphs. A digraph group is a group defined by non-empty presentation with the property that each relator is of the form \(R(x, y)\), where x and y are distinct generators and \(R(\cdot , \cdot )\) is determined by some fixed cyclically reduced word \(R(a, b)\) that involves both a and b. Associated with each such presentation is a digraph whose vertices correspond to the generators and whose arcs correspond to the relators. In this article, we consider digraph groups for strong digraphs that are digon-free and triangle-free. We classify when the digraph group is finite and show that in these cases it is cyclic, giving its order. We apply this result to the Cayley digraph of the generalized quaternion group, to circulant digraphs, and to Cartesian and direct products of strong digraphs. |
| Author | Williams, Gerald Cihan, Mehmet Sefa |
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| Copyright | The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society. This work is licensed under the Creative Commons Attribution License This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| Keywords | Digraph group digraph products 05C20 Cayley digraph 05C40 20F05 05C25 05C38 strong digraph circulant digraph |
| Language | English |
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| References | 2010; 23 1963; 14 1987; 36 1979; 19 1966; 14 2024; 228 1967; 2 1962; 14 2022; 41 2020; 224 2024; 27 1959; 10 2016; 284 1976; 19 |
| References_xml | – volume: 284 start-page: 507 issue: 1–2 year: 2016 end-page: 535 article-title: Efficient finite groups arising in the study of relative asphericity publication-title: Math. Z. – volume: 19 start-page: 247 issue: 2 year: 1976 end-page: 248 article-title: On a group presentation due to Fox publication-title: Canad. Math. Bull. – volume: 23 start-page: 796 issue: 7 year: 2010 end-page: 800 article-title: Cyclic arc-connectivity in a Cartesian product digraph publication-title: Appl. Math. Lett. – volume: 228 start-page: 107499 issue: 4 year: 2024 article-title: Finite groups defined by presentations in which each defining relator involves exactly two generators publication-title: J. Pure Appl. Algebra – volume: 14 start-page: 600 year: 1963 end-page: 606 article-title: On the product of directed graphs publication-title: Proc. Amer. Math. Soc. – volume: 19 start-page: 59 issue: 1 year: 1979 end-page: 61 article-title: A new class of $3$ -generator finite groups of deficiency zero publication-title: J. Lond. Math. Soc. (2) – volume: 14 start-page: 250 year: 1966 end-page: 254 article-title: Connectedness of products of two directed graphs publication-title: SIAM J. Appl. Math. – volume: 41 start-page: 31 year: 2022 end-page: 35 article-title: Digraph groups corresponding to digraphs with one more vertex than arcs publication-title: Avrupa Bilim ve Teknoloji Dergisi – volume: 27 start-page: 549 issue: 3 year: 2024 end-page: 594 article-title: Structure of the Macdonald groups in one parameter publication-title: J. Group Theory – volume: 36 start-page: 245 issue: 2 year: 1987 end-page: 256 article-title: Groups with presentations in which each defining relator involves exactly two generators publication-title: J. Lond. Math. Soc. (2) – volume: 14 start-page: 602 year: 1962 end-page: 613 article-title: On a class of finitely presented groups publication-title: Canad. J. Math. – volume: 10 start-page: 409 year: 1959 end-page: 418 article-title: Einige endliche Gruppen mit drei Erzeugenden und drei Relationen publication-title: Arch. Math. – volume: 2 start-page: 393 year: 1967 article-title: Research problem 2-10 publication-title: J. Combin. Theory – volume: 224 start-page: 106342 issue: 8 year: 2020 article-title: A class of digraph groups defined by balanced presentations publication-title: J. Pure Appl. Algebra |
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| Snippet | A digraph group is a group defined by non-empty presentation with the property that each relator is of the form
$R(x, y)$
, where x and y are distinct... A digraph group is a group defined by non-empty presentation with the property that each relator is of the form $R(x, y)$ , where x and y are distinct... A digraph group is a group defined by non-empty presentation with the property that each relator is of the form \(R(x, y)\), where x and y are distinct... |
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| Title | Strong digraph groups |
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