Classification of Subsets of the Projective Line of Order Thirty Two and its Partitioned into Distinct Subsets
The aim of this paper is to find the inequivalent k-sets in the finite projective line of order thirty-two, PG(1,32). The number of projectively distinct 4-set is five and all of them are of type N(neither harmonic nor equianharmonic). The k-sets, k=4,…,11 have been done, where the number of projec...
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University of Baghdad
2025-04-01
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| Series: | Ibn Al-Haitham Journal for Pure and Applied Sciences |
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| Online Access: | https://jih.uobaghdad.edu.iq/index.php/j/article/view/3705 |
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| author | zainab abbas Emad Al-Zangana Mohammed M. Ali Al-Shamiri |
| author_facet | zainab abbas Emad Al-Zangana Mohammed M. Ali Al-Shamiri |
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The aim of this paper is to find the inequivalent k-sets in the finite projective line of order thirty-two, PG(1,32). The number of projectively distinct 4-set is five and all of them are of type N(neither harmonic nor equianharmonic). The k-sets, k=4,…,11 have been done, where the number of projectively distinct are 5,11,53,148,481,1240,2964,6049, respectively. The k-sets k=12,..,17 classified depending on the projectively distinct 11-sets whose have non-trivial subgroups only, where the numbers of projectively distinct are 493,5077,2583,288,2412,697. The stabilizer group of each k-sets is computed. The kind of groups that computed for the k-sets are I, Z_2, Z_3, V_4, S_3, Z_2×Z_2×Z_2, Z_2×Z_2×Z_2×Z_2 and the large group is the dihedral group of order eleven appears when k is equal to eleven. Also, the projective line PG(1,32) is partitioned into three distinct 11-sets such that two of them are projectively equivalent, and into eight 4-sets of types N_1, N_2, N_3,. N_4, N_5, and into eight 4-sets four of them of type N_3, N_4.
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| institution | OA Journals |
| issn | 1609-4042 2521-3407 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | University of Baghdad |
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| series | Ibn Al-Haitham Journal for Pure and Applied Sciences |
| spelling | doaj-art-9a9081c9c0f548428d4c94a02e4ffbea2025-08-20T02:12:19ZengUniversity of BaghdadIbn Al-Haitham Journal for Pure and Applied Sciences1609-40422521-34072025-04-0138210.30526/38.2.3705Classification of Subsets of the Projective Line of Order Thirty Two and its Partitioned into Distinct Subsetszainab abbas0https://orcid.org/0009-0005-8511-449XEmad Al-Zangana1https://orcid.org/0000-0001-6415-1930Mohammed M. Ali Al-Shamiri 2https://orcid.org/0000-0001-6124-9043Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq.Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq.Department of Mathematics, College of Sciences and Arts, Muhyil Assir, King Khalid University, Muhyil, Saudi Arabia. The aim of this paper is to find the inequivalent k-sets in the finite projective line of order thirty-two, PG(1,32). The number of projectively distinct 4-set is five and all of them are of type N(neither harmonic nor equianharmonic). The k-sets, k=4,…,11 have been done, where the number of projectively distinct are 5,11,53,148,481,1240,2964,6049, respectively. The k-sets k=12,..,17 classified depending on the projectively distinct 11-sets whose have non-trivial subgroups only, where the numbers of projectively distinct are 493,5077,2583,288,2412,697. The stabilizer group of each k-sets is computed. The kind of groups that computed for the k-sets are I, Z_2, Z_3, V_4, S_3, Z_2×Z_2×Z_2, Z_2×Z_2×Z_2×Z_2 and the large group is the dihedral group of order eleven appears when k is equal to eleven. Also, the projective line PG(1,32) is partitioned into three distinct 11-sets such that two of them are projectively equivalent, and into eight 4-sets of types N_1, N_2, N_3,. N_4, N_5, and into eight 4-sets four of them of type N_3, N_4. https://jih.uobaghdad.edu.iq/index.php/j/article/view/3705Cross-ratio, Finite field, Partition of sets, Projective line |
| spellingShingle | zainab abbas Emad Al-Zangana Mohammed M. Ali Al-Shamiri Classification of Subsets of the Projective Line of Order Thirty Two and its Partitioned into Distinct Subsets Ibn Al-Haitham Journal for Pure and Applied Sciences Cross-ratio, Finite field, Partition of sets, Projective line |
| title | Classification of Subsets of the Projective Line of Order Thirty Two and its Partitioned into Distinct Subsets |
| title_full | Classification of Subsets of the Projective Line of Order Thirty Two and its Partitioned into Distinct Subsets |
| title_fullStr | Classification of Subsets of the Projective Line of Order Thirty Two and its Partitioned into Distinct Subsets |
| title_full_unstemmed | Classification of Subsets of the Projective Line of Order Thirty Two and its Partitioned into Distinct Subsets |
| title_short | Classification of Subsets of the Projective Line of Order Thirty Two and its Partitioned into Distinct Subsets |
| title_sort | classification of subsets of the projective line of order thirty two and its partitioned into distinct subsets |
| topic | Cross-ratio, Finite field, Partition of sets, Projective line |
| url | https://jih.uobaghdad.edu.iq/index.php/j/article/view/3705 |
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