Classification of Subsets of the Projective Line of Order Thirty Two and its Partitioned into Distinct Subsets - Abbas, Zainab; Al-Zangana, Emad; Ali Al-Shamiri , Mohammed M. (Ibn AL-Haitham Journal For Pure and Applied Sciences, 2025)
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.