On invertor elements and finitely generated subgroups of groups acting on trees with inversions
An element of a group acting on a graph is called invertor if it transfers an edge of the graph to its inverse. In this paper, we show that if G is a group acting on a tree X with inversions such that G does not fix any element of X, then an element g of G is invertor if and only if g is not in any...
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| Format: | Article |
| Language: | English |
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Wiley
2000-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171200002969 |
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| author | R. M. S. Mahmood M. I. Khanfar |
| author_facet | R. M. S. Mahmood M. I. Khanfar |
| author_sort | R. M. S. Mahmood |
| collection | DOAJ |
| description | An element of a group acting on a graph is called invertor if it
transfers an edge of the graph to its inverse. In this paper, we
show that if G is a group acting on a tree X with inversions
such that G does not fix any element of X, then an element g of G is invertor if and only if g is not in any vertex
stabilizer of G and g2 is in an edge stabilizer of G. Moreover, if H is a finitely generated subgroup of G, then H contains an invertor element or some conjugate of H contains a
cyclically reduced element of length at least one on which H is
not in any vertex stabilizer of G, or H is in a vertex
stabilizer of G. |
| format | Article |
| id | doaj-art-bf2c74865acb404ea99fe37fce144ddd |
| institution | DOAJ |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2000-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-bf2c74865acb404ea99fe37fce144ddd2025-08-20T03:23:07ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252000-01-0123958559510.1155/S0161171200002969On invertor elements and finitely generated subgroups of groups acting on trees with inversionsR. M. S. Mahmood0M. I. Khanfar1Ajman University of Science and Technology, Abu Dhabi, United Arab EmiratesDepartment of Mathematics, Yarmouk University, Irbid, JordanAn element of a group acting on a graph is called invertor if it transfers an edge of the graph to its inverse. In this paper, we show that if G is a group acting on a tree X with inversions such that G does not fix any element of X, then an element g of G is invertor if and only if g is not in any vertex stabilizer of G and g2 is in an edge stabilizer of G. Moreover, if H is a finitely generated subgroup of G, then H contains an invertor element or some conjugate of H contains a cyclically reduced element of length at least one on which H is not in any vertex stabilizer of G, or H is in a vertex stabilizer of G.http://dx.doi.org/10.1155/S0161171200002969Groups acting on treesinvertor elementsfinitely generated subgroups. |
| spellingShingle | R. M. S. Mahmood M. I. Khanfar On invertor elements and finitely generated subgroups of groups acting on trees with inversions International Journal of Mathematics and Mathematical Sciences Groups acting on trees invertor elements finitely generated subgroups. |
| title | On invertor elements and finitely generated subgroups of groups acting on trees with inversions |
| title_full | On invertor elements and finitely generated subgroups of groups acting on trees with inversions |
| title_fullStr | On invertor elements and finitely generated subgroups of groups acting on trees with inversions |
| title_full_unstemmed | On invertor elements and finitely generated subgroups of groups acting on trees with inversions |
| title_short | On invertor elements and finitely generated subgroups of groups acting on trees with inversions |
| title_sort | on invertor elements and finitely generated subgroups of groups acting on trees with inversions |
| topic | Groups acting on trees invertor elements finitely generated subgroups. |
| url | http://dx.doi.org/10.1155/S0161171200002969 |
| work_keys_str_mv | AT rmsmahmood oninvertorelementsandfinitelygeneratedsubgroupsofgroupsactingontreeswithinversions AT mikhanfar oninvertorelementsandfinitelygeneratedsubgroupsofgroupsactingontreeswithinversions |