Classification of Subsets of the Projective Line of Order Thirty Two and its Partitioned into Distinct Subsets

Zainab Abbas; Emad  Al-Zangana; Mohammed M.  Ali Al-Shamiri

doi:10.30526/38.2.3705

Authors

Zainab Abbas Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq. https://orcid.org/0009-0005-8511-449X
Emad Al-Zangana Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq. https://orcid.org/0000-0001-6415-1930
Mohammed M. Ali Al-Shamiri Department of Mathematics, College of Sciences and Arts, Muhyil Assir, King Khalid University, Muhyil, Saudi Arabia. https://orcid.org/0000-0001-6124-9043

DOI:

https://doi.org/10.30526/38.2.3705

Keywords:

Cross-ratio, Finite field, Partition of sets, Projective line

Abstract

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.

Author Biographies

Zainab Abbas, Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq.

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Emad Al-Zangana, Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq.

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Mohammed M. Ali Al-Shamiri , Department of Mathematics, College of Sciences and Arts, Muhyil Assir, King Khalid University, Muhyil, Saudi Arabia.

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