348 

© 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

 
 

 

 

Classification of Subsets of the Projective Line of Order Thirty-Two and 

its Partitioning into Distinct Subsets 

 

Zainab Abbas Khalaf 1* , Emad Bakr Al-Zangana 2  and Mohammed M. Ali 

Al-Shamiri 3   
1,2Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq. 

3Department of Mathematics, College of Science and Arts, Muhyil Assir, King Khalid University, 

Muhyil, Saudi Arabia. 
*Corresponding Author. 

 

Received: 13 September 2023            Accepted: 19 February 2024             Published: 20 April 2025  

doi.org/10.30526/38.2.3705 

 

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, 𝑍2, 𝑍3, 𝑉4, 𝑆3, 𝑍2 × 𝑍2 × 𝑍2, 𝑍2 × 𝑍2 × 𝑍2 × 𝑍2 

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 𝑁1, 𝑁2, 𝑁3,. 𝑁4, 𝑁5, and into 

eight 4-sets four of them of type 𝑁3, 𝑁4. 

Keywords: Cross-ratio, Finite field, Partition of sets, Projective line. 

 

1. Introduction 

     Let 𝐹𝑞 = {∞, 0,1, 𝜔, 𝜔2, … 𝜔𝑞−2} be a finite field generated by 𝜔.  In 𝑃𝐺(1, 𝑞), a 𝑘-set 

can be formed by adding one point from the other 𝑞 − 𝑘 + 2 points to any (𝑘 − 1)-set. From 

the Fundamental Theorem of Projective Geometry, any three points on a line are projectively 

equivalent. See [1. Ch. 6]. The points of 𝑃𝐺(1, 𝑞) are 𝑃(𝑥0, 𝑥1), 𝑥0 and 𝑥1 ∈ 𝐹𝑞 but not both 

zero. Each point 𝑃(𝑥0, 𝑥1), with 𝑥1 ≠ 0 is determined by the non-homogeneous coordinate 

𝑥0 𝑥1⁄ ; the coordinate for point 𝑃(1,0) is ∞. Then, the point of 𝑃𝐺(1, 𝑞) can be represented 

by the set 𝐹𝑞 ∪ {∞}. A projectivity 𝜑 = 𝑀(𝐴) of 𝑃𝐺(1, 𝑞) is given by 𝑌 =  𝑋𝐴, where 𝑋 = 

(𝑥0, 𝑥1), 𝑌 = (𝑦0, 𝑦1) and 𝐴 = [
𝑎 𝑏
𝑐 𝑑

]. Let 𝑠 = 𝑦0 𝑦1⁄  and 𝑡 = 𝑥0 𝑥1⁄ ; then 𝑠 =

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0009-0005-8511-449X
mailto:zaynab.ab98@gmail.com
https://orcid.org/0000-0001-6415-1930
mailto:e.b.abdulkrareem@uomustansiriyah.edu.iq
https://orcid.org/0000-0001-6124-9043
mailto:Mal-shamiri@kku.edu.sa


 

IHJPAS. 2025,38(2) 

349 
 

(𝑎𝑡 + 𝑐 𝑏𝑡 + 𝑑)⁄ . If 𝑄𝑖 = 𝑃𝑖𝐴  for 𝑖 =2,3,4 and 𝑃𝑖  , 𝑄𝑖 have the respective coordinates 𝑡𝑖 and 

𝑠𝑖, then  𝜑 is given by  

(𝑠−𝑠3)(𝑠2−𝑠4)

(𝑠−𝑠4)(𝑠2−𝑠3)
=

(𝑡−𝑡3)(𝑡2−𝑡4)

(𝑡−𝑡4)(𝑡2−𝑡3)
. 

In (1), the classification  of the projective lines over Galois field of order 𝑞 = 2,3,4,5,7,8,9 

are given. In (2)the author did in his thesis, a classification of 𝑃𝐺(1,11),  and in (3) the author 

did in his thesis a classification of 𝑃𝐺(1,13). In (4), the authors classified the 𝑘-sets in 

projective line of order twenty-seven with partition of the space into five 4-sets, one of type 

𝐸 (equianharmonic) and four of type 𝑁 (neither harmonic nor equianharmonic), and the full 

classification of 𝑃𝐺(1, 𝑞), 𝑞 = 19,23,25 with its application into error-correcting codes have 

been done as mentioned in the sources of (4).  In (5) the authors studied the geometry of line 

in 𝑃𝐺(1,17) rise up to error-correcting code. In (6) the author gave the full classification of 

inequivalent 𝑘-sets in 𝑃𝐺(1,16),  and for some 𝑘 in 𝑃𝐺(1,29) and in 𝑃𝐺(1,31) as in (7) and 

sources therein. The study of finite dimensional finite projective space has been done by 

many authors for specific field 𝐹𝑞 as appears in the sources (8-28).  

Definition 1. (1): The cross-ratio of four ordered distinct points 𝑃1, 𝑃2, 𝑃3, 𝑃4 with coordinates 

𝑡1, 𝑡2, 𝑡3, 𝑡4 is 

𝜆 = {𝑃1, 𝑃2; 𝑃3, 𝑃4}  = {𝑡1, 𝑡2; 𝑡3, 𝑡4} =
(𝑡1−𝑡3)(𝑡2−𝑡4)

(𝑡1−𝑡4)(𝑡2−𝑡3)
 . 

The cross-ratio has property that  

𝜆={𝑡1, 𝑡2; 𝑡3, 𝑡4} = {𝑡2, 𝑡1; 𝑡4, 𝑡3} = {𝑡3, 𝑡4; 𝑡1, 𝑡2} = {𝑡4, 𝑡3; 𝑡2, 𝑡1}. 

So {𝑃1, 𝑃2; 𝑃3, 𝑃4} is invariant under a projective group of order four (Klein Group) 𝑉4. 

Thus, under all permutations of {𝑃1, 𝑃2; 𝑃3, 𝑃4}, the cross-ratio take just the six values 

λ,   1 λ,   1 − λ,   1 (1 − λ),   (λ − 1) λ,   λ (λ − 1)⁄⁄⁄⁄ . 

Also, {𝑡1, 𝑡2; 𝑡3, 𝑡4} takes the values ∞, 0 or 1 if and only if two of the 𝑡𝑖 are equal(1). 

Definition 2. (1): The 4-set is called harmonic, denoted by 𝐻, if the cross-ratio are −1,2,1/2, 

equianharmonic, denoted by 𝐸, if λ = 1 (1 − λ)⁄  or λ = (λ − 1) λ⁄  and neither harmonic nor 

equianharmonic, denoted by 𝑁, if the cross-ratio another value. 

 Clear the characteristic of 𝐹𝑞 is 2, so there are no harmonic 4-set.  When  𝑝 = 3, then 𝜆 =

−1 = 2 = 1 2⁄ . The cross-ratio of type 𝐸 exist if 𝑞 ≡ 1 or 0 (mod 3).  

Definition 3. Let 𝜌1 and 𝜌2 be two projective spaces of 𝑛-dimension. A projectivity 𝜑: 𝜌1 ⟶

𝜌2 is a bijection given by a non-singular matrix 𝐴 such that 𝑃(𝑋′) = 𝑃(𝑋)𝜑 if and only if 

𝑡𝑋′ = 𝑋𝐴, where 𝑡 ∈ 𝐹𝑞\ {0}. Write 𝜑 = 𝑀(𝐴), then 𝜑 = 𝑀(𝜆𝐴) for any 𝜆 ∈ 𝐹𝑞\ {0}. 

To determine a projectivity (non-singular 2 × 2 matrix) on the projective line it enough to 

have three distinct points.  

 

2. Materials and Methods 

2.1.The Projective Line of Order 32  

 In 𝑃𝐺(1,32), the projective line over Galois field of order 32, there are 33 points. The points 

of 𝑃𝐺(1,32) are 𝐹32 ∪ {∞} ={∞,0,1,𝜔,…,𝜔30}. 

The polynomial function 𝑓(𝑥) = 𝑥2 + 𝜔6𝑥 + 𝜔  is primitive over 𝐹32, then 33 points of 

𝑃𝐺(1,32)can be generated by non-singular matrix; 𝐴 = 𝐶(𝑓) = [[0 1], [𝜔 𝜔6]], such 

that 𝑃(𝑖) = (1,0)𝐴𝑖, 𝑖 = 0, … ,32 as in Table 1. 

 

 

 



 

IHJPAS. 2025,38(2) 

350 
 

 

Table 1. The points of 𝑃𝐺(1,32). 

𝑃(0) = [1 ,0 ] 𝑃(1) = [0 ,1 ] 𝑃(2) = [𝜔26 ,1 ] 

𝑃(3) = [𝜔18 ,1 ] 𝑃(4) = [𝜔3 ,1 ] 𝑃(5) = [1 ,1 ] 

𝑃(6) = [𝜔5 ,1 ] 𝑃(7) = [𝜔9 ,1 ] 𝑃(8) = [𝜔28 ,1 ] 

𝑃(9) = [𝜔19 ,1 ] 𝑃(10) = [𝜔12 ,1 ] 𝑃(11) = [𝜔30 ,1 ] 

𝑃(12) = [𝜔11 ,1 ] 𝑃(13) = [𝜔24 ,1 ] 𝑃(14) = [𝜔25 ,1 ] 

𝑃(15) = [𝜔15 ,1 ] 𝑃(16) = [𝜔10 ,1 ] 𝑃(17) = [𝜔16 ,1 ] 

𝑃(18) = [𝜔22 ,1 ] 𝑃(19) = [𝜔17 ,1 ] 𝑃(20) = [𝜔7 ,1 ] 

𝑃(21) = [𝜔8 ,1 ] 𝑃(22) = [𝜔21 ,1 ] 𝑃(23) = [𝜔2 ,1 ] 

𝑃(24) = [𝜔20 ,1 ] 𝑃(25) = [𝜔13 ,1 ] 𝑃(26) = [𝜔4 ,1 ] 

𝑃(27) = [𝜔23 ,1 ]  𝑃(28) = [𝜔27 ,1 ] 𝑃(29) = [𝜔 ,1 ] 

𝑃(30) = [𝜔29 ,1 ]                       𝑃(31) = [𝜔14 ,1 ] 𝑃(32) = [𝜔6 ,1 ] 

 

3. Results and  Discussion 

     This section includes the classification’s results of the projective line 𝑃𝐺(1,32) into 𝑘-

sets, where 𝑘 = 4, … ,17. 

3.1. The 4-sets 

Let 𝜉 be all different 3-sets in 𝑃𝐺(1,32). Then the order of 𝜉 is |ξ| = 33 ∙ 32 ∙ 31 = 32736. 

But as mentioned in Section 3, any three distinct points on a line are projectively equivalent, 

so we can fixed the 3-set, 𝒪 = {∞, 0,1}  to construct (3 + 𝑖)-set, 𝑖 = 0,1, … ,
𝑞−5

2
, 𝑞 > 5  if 𝑞 

odd and 𝑖 = 0,1, … ,
𝑞−4

2
, 𝑞 > 4 if 𝑞 even.  

A 4-set is constructed by adding to 𝒪 = {∞, 0,1} one point from the complement of 𝒪. 

Let 𝒮 be the set of all different 4-set in 𝑃𝐺(1,32). Then 𝒮 has order  |𝒮| = (33
4

) = 40920. 

A 4-set of type 𝐻 and 𝐸 when 𝑞 = 25 does not exist but the 4-set of type 𝑁 has been divided 

into 5 classes. 

𝑁1 ∋ {∞, 0,1, 𝑎}, 𝑎 = {𝜔, 𝜔13, 𝜔14, 𝜔17, 𝜔18, 𝜔30}; 

𝑁2 ∋ {∞, 0,1, 𝑏}, 𝑏 = {𝜔2, 𝜔3, 𝜔5, 𝜔26, 𝜔28, 𝜔29}; 

𝑁3 ∋ {∞, 0,1, 𝑐}, 𝑐 = {𝜔4, 𝜔6, 𝜔10, 𝜔21, 𝜔25, 𝜔27}; 

𝑁4 ∋ {∞, 0,1, 𝑑}, 𝑑 = {𝜔7, 𝜔9, 𝜔15, 𝜔16, 𝜔22, 𝜔24}; 

𝑁5 ∋ {∞, 0,1, 𝑒}, 𝑒 = {𝜔8, 𝜔11, 𝜔12, 𝜔19, 𝜔20, 𝜔23}. 

Since any two 4-sets with same cross-ratio are projectively equivalent, so each class 𝑁𝑖, 𝑖 =

1, … 5 is projectively unique. Then among the 40920 of 4-sets there are only five projectively 

distinct 4-sets, which are given in Table 2 with its stabilizer group type denoted by SG. 

 

Table 2. The Inequivalent 4-set 

. 4-set Symbol 

𝑉4 =< 𝜔 𝑡⁄ , 𝑡 + 𝜔 𝑡 + 1⁄ > {∞, 0,1, 𝜔} 𝒯1 

𝑉4 =< 𝜔2 𝑡⁄ , 𝑡 + 𝜔2 𝑡 + 1⁄ > {∞, 0,1, 𝜔2} 𝒯2 

𝑉4 =< 𝜔4 𝑡⁄ , 𝑡 + 𝜔4 𝑡 + 1⁄ > {∞, 0,1, 𝜔4} 𝒯3 

𝑉4 =< 𝜔7 𝑡⁄ , 𝑡 + 𝜔7 𝑡 + 1⁄ > {∞, 0,1, 𝜔7} 𝒯4 

𝑉4 =< 𝜔8 𝑡⁄ , 𝑡 + 𝜔8 𝑡 + 1⁄ > {∞, 0,1, 𝜔8} 𝒯5 

 

 

 

 



 

IHJPAS. 2025,38(2) 

351 
 

Remark 4. 

(i) To reduce the number of constructed (3 + 𝑖)-sets, we will use idea of group action to 

partition 𝑃𝐺(1,32) into distinct orbits and take the first point from each orbit to do the 

extension of  (3 + 𝑖)-sets.  

(ii) The GAP program (29) is used to find the action groups, to find the stabilizer group of 

each (3 + 𝑖)-set, and to run the algorithm (see (2)) which is used find the non-equivalents 

(3 + 𝑖)-sets. 

(iii) To know the kind of stabilizer group of order between 4 and 32 from its structure the 

reference (30) is used.  

3.2. The 5-sets  

The projective group 𝐺𝒯𝑖
 acts on 𝒯𝑖

𝑐 from the right and splitting it into 5 orbits, four of them 

of order four and one of them singleton set. Then 5-set constructed by adding one point from 

each different orbit as in Table 3. 

 

Table 3. Partition of 𝒯𝑖
𝑐 by the projectivities of 4-set 

Partition of 𝓣𝒊
𝒄 𝓣𝒊 

{𝜔2, 𝜔30, 𝜔14, 𝜔18}, {𝜔3, 𝜔29, 𝜔8, 𝜔24}, {𝜔4, 𝜔28, 𝜔20, 𝜔12}, {𝜔5, 𝜔27, 𝜔9, 𝜔23}, 

{𝜔6, 𝜔26, 𝜔7, 𝜔25}, {𝜔10, 𝜔22, 𝜔13, 𝜔19}, {𝜔11, 𝜔21, 𝜔17, 𝜔15}, {𝜔16} 

𝒯1 

{𝜔5}, {𝜔3, 𝜔30, 𝜔22, 𝜔11}, {𝜔4, 𝜔29, 𝜔28, 𝜔5}, {𝜔6, 𝜔27, 𝜔16, 𝜔17}, 

{𝜔7, 𝜔26, 𝜔13, 𝜔20}, {𝜔8, 𝜔25, 𝜔9, 𝜔24}, {𝜔10, 𝜔23, 𝜔18, 𝜔15}, {𝜔12, 𝜔21, 𝜔14, 𝜔19} 

𝒯2 

{𝜔, 𝜔3, 𝜔12, 𝜔23}, {𝜔2}, {𝜔5, 𝜔30, 𝜔20, 𝜔15}, {𝜔6, 𝜔29, 𝜔13, 𝜔22}, 

{𝜔7, 𝜔28, 𝜔11, 𝜔24}, {𝜔8, 𝜔27, 𝜔25, 𝜔10}, {𝜔9, 𝜔26, 𝜔21, 𝜔14}, {𝜔16, 𝜔19, 𝜔18, 𝜔17} 

𝒯3 

{𝜔, 𝜔6, 𝜔10, 𝜔28}, {𝜔2, 𝜔5, 𝜔30, 𝜔8}, {𝜔3, 𝜔4, 𝜔15, 𝜔23}, {𝜔9, 𝜔29, 𝜔27, 𝜔11}, 

{𝜔12, 𝜔26, 𝜔17, 𝜔21}, {𝜔13, 𝜔25, 𝜔20, 𝜔18}, {𝜔14, 𝜔24, 𝜔16, 𝜔22}, {𝜔19} 

𝒯4 

{𝜔, 𝜔7, 𝜔5, 𝜔3}, {𝜔2, 𝜔6, 𝜔24, 𝜔15}, {𝜔4}, {𝜔9, 𝜔30, 𝜔10, 𝜔29}, 

{𝜔11, 𝜔28, 𝜔18, 𝜔21}, {𝜔12, 𝜔27, 𝜔26, 𝜔13}, {𝜔14, 𝜔25, 𝜔22, 𝜔17}, {𝜔16, 𝜔23, 𝜔19, 𝜔20} 

𝒯5 

During the research, the sequence of 𝑖1, 𝑖2, … , 𝑖𝑛 refer to type of (𝑛 − 1)-sets in 𝑛-set. 

Theorem 5: In 𝑃𝐺(1,32), there are 11 projectively inequivalent 5-sets, summarized in Table 

4. 

Table 4. Inequivalent 5-set 

SG Type of 5-set 5-set Symbol 

𝑉4 =< 𝜔2 𝑡⁄ , 𝑡 + 𝜔2 𝑡 + 1⁄ > 1 1 1 1 2 {∞, 0,1, 𝜔, 𝜔2} 𝑓1 

I 1 2 2 4 5 {∞, 0,1, 𝜔, 𝜔3} 𝑓2 

I 1 2 3 5 5 {∞, 0,1, 𝜔, 𝜔4} 𝑓3 

I 1 2 3 4 5 {∞, 0,1, 𝜔, 𝜔5} 𝑓4 

I 1 2 3 3 4 {∞, 0,1, 𝜔, 𝜔6} 𝑓5 

I 1 1 3 4 5 {∞, 0,1, 𝜔, 𝜔10} 𝑓6 

𝑉4 = < 𝜔 𝑡⁄ , 𝑡 + 𝜔 𝑡 + 1⁄ > 1 4 4 4 4 {∞, 0,1, 𝜔, 𝜔16} 𝑓7 

𝑉4 =< 𝜔4 𝑡⁄ , 𝑡 + 𝜔4 𝑡 + 1⁄ > 2 2 2 2 3 {∞, 0,1, 𝜔2, 𝜔4} 𝑓8 

I 2 3 4 4 5 {∞, 0,1, 𝜔2, 𝜔8} 𝑓9 

𝑉4 = < 𝜔8 𝑡⁄ , 𝑡 + 𝜔8 𝑡 + 1⁄ > 3 3 3 3 5 {∞, 0,1, 𝜔4, 𝜔8} 𝑓10 

𝑉4 =< 𝜔7 𝑡⁄ , 𝑡 + 𝜔7 𝑡 + 1⁄ > 4 5 5 5 5 {∞, 0,1, 𝜔7, 𝜔19} 𝑓11 



 

IHJPAS. 2025,38(2) 

352 
 

 

 

 

 

 

 

 

 

 

Table 5. Inequivalent 6-sets 
SG Type of 5-set 6-set Symbol 

𝑍2 =< 𝜔3 𝑡⁄ > 1 1 2 2 6 6 𝑓1 ∪ {𝜔3} ℎ1 

I 1 2 3 3 4 8 𝑓1 ∪ {𝜔4} ℎ2 

I 1 4 5 5 6 7 𝑓1 ∪ {𝜔6} ℎ3 

I 1 2 3 5 5 6 𝑓1 ∪ {𝜔7} ℎ4 

I 1 2 2 4 5 9 𝑓1 ∪ {𝜔8} ℎ5 

𝑍2 =< 𝑡𝜔19 + 𝜔29 𝑡𝜔17 + 𝜔19⁄ > 1 1 4 4 6 6 𝑓1 ∪ {𝜔10} ℎ6 

𝑍2 =< 𝑡 + 𝜔2 𝑡 + 1⁄ > 1 1 3 3 6 6 𝑓1 ∪ {𝜔12} ℎ7 

𝑍2 =< 𝜔4 𝑡⁄ > 2 2 3 3 11 11 𝑓2 ∪ {𝜔4} ℎ8 

𝑍2 =< 𝑡𝜔11 + 𝜔14 𝑡𝜔6 + 𝜔11⁄ > 2 2 4 4 8 8 𝑓2 ∪ {𝜔5} ℎ9 

𝑍2 =< 𝑡𝜔6 + 𝜔12 𝑡𝜔3 + 𝜔6⁄ > 2 2 5 5 8 8 𝑓2 ∪ {𝜔6} ℎ10 

𝑍2 =< 𝑡𝜔13 + 𝜔14 𝑡𝜔3 + 𝜔6⁄ > 2 2 5 5 9 9 𝑓2 ∪ {𝜔7} ℎ11 

𝑆3 =< 𝑡 + 𝜔 𝑡 + 1 + 𝜔8⁄ , 𝑡𝜔3 + 𝜔6 𝑡⁄ > 2 2 2 2 2 2 𝑓2 ∪ {𝜔8} ℎ12 

𝑍2 =< 𝑡 + 𝜔9 𝑡 + 1⁄ > 2 2 4 4 9 9 𝑓2 ∪ {𝜔9} ℎ13 

I 2 3 4 5 6 9 𝑓2 ∪ {𝜔10} ℎ14 

I 2 3 4 6 6 7 𝑓2 ∪ {𝜔11} ℎ15 

𝑍2 =< 𝑡𝜔20 + 𝜔23 𝑡𝜔19 + 𝜔20⁄ > 2 2 3 3 9 9 𝑓2 ∪ {𝜔12} ℎ16 

I 2 3 4 5 6 9 𝑓2 ∪ {𝜔13} ℎ17 

𝑍2 =< 𝜔3(𝑡 + 1) 𝑡 + 𝜔3⁄ > 2 2 3 3 6 6 𝑓2 ∪ {𝜔15} ℎ18 

I 2 2 4 7 9 11 𝑓2 ∪ {𝜔16} ℎ19 

𝑍2 =< 𝑡 + 𝜔3 𝑡 + 1⁄ > 2 2 4 4 6 6  𝑓2 ∪ {𝜔19} ℎ20 

I 2 3 3 4 9 10 𝑓2 ∪ {𝜔20} ℎ21 

I 2 2 5 6 7 9 𝑓2 ∪ {𝜔21} ℎ22 

I 2 3 6 6 9 11 𝑓2 ∪ {𝜔22} ℎ23 

𝑍2 =< 𝑡𝜔11 + 𝜔12 𝑡𝜔8 + 𝜔11⁄ > 2 2 3 3 4 4  𝑓2 ∪ {𝜔23} ℎ24 

𝑍2 =< 𝜔(𝑡 + 1) 𝑡 + 𝜔⁄ > 2 2 4 4 5 5 𝑓2 ∪ {𝜔24} ℎ25 

I 2 3 4 5 6 9 𝑓2 ∪ {𝜔25} ℎ26 

I 2 3 3 5 8 9 𝑓2 ∪ {𝜔28} ℎ27 

𝑍2 =< 𝜔 𝑡⁄ > 2 2 8 8 9 9 𝑓3 ∪ {𝜔29} ℎ28 

𝑍2 =< 𝜔5 𝑡⁄ > 3 3 4 4 11 11 𝑓3 ∪ {𝜔5} ℎ29 

I 3 4 5 5 6 8 𝑓3 ∪ {𝜔6} ℎ30 

I 3 4 6 6 10 11 𝑓3 ∪ {𝜔10} ℎ31 

𝑆3 =< 𝑡 + 𝜔11 𝑡 + 1⁄ , 𝑡𝜔12 + 𝜔24 𝑡⁄ > 3 3 3 3 3 3 𝑓3 ∪ {𝜔12} ℎ32 

I 3 5 6 9 9 10 𝑓3 ∪ {𝜔13} ℎ33 

𝑍2 =< 𝑡𝜔14 + 𝜔18 𝑡𝜔30 + 𝜔14⁄ > 3 3 4 4 6 6 𝑓3 ∪ {𝜔15} ℎ34 

𝑍2 =< 𝑡𝜔30 + 𝜔21 𝑡𝜔26 + 𝜔30⁄ > 3 3 5 5 6 6  𝑓3 ∪ {𝜔22} ℎ35 

𝑍2 =< 𝜔4(𝑡 + 1) 𝑡 + 𝜔4⁄ > 3 3 4 4 9 9  𝑓3 ∪ {𝜔23} ℎ36 

𝑍2 =< 𝑡𝜔30 + 𝜔24 𝑡𝜔29 + 𝜔30⁄ > 3 3 5 5 10 10  𝑓3 ∪ {𝜔25} ℎ37 

𝑍2 =< 𝑡 + 𝜔26 𝑡 + 1⁄ > 3 3 4 4 5 5 𝑓3 ∪ {𝜔26} ℎ38 

𝑍2 =< 𝜔 𝑡⁄ > 3 3 9 9 11 11 𝑓3 ∪ {𝜔28} ℎ39 

𝑍2 =< 𝜔6 𝑡⁄ > 4 4 5 5 9 9 𝑓4 ∪ {𝜔6} ℎ40 

𝑍2 =< 𝑡 + 𝜔 𝑡 + 1⁄ > 4 4 5 5 10 10 𝑓4 ∪ {𝜔9} ℎ41 

I 4 5 6 9 9 11 𝑓4 ∪ {𝜔15} ℎ42 

𝑍2 =< 𝑡𝜔25 + 𝜔30 𝑡𝜔24 + 𝜔25⁄ > 4 4 7 7 9 9 𝑓4 ∪ {𝜔16} ℎ43 

𝑍2 =< 𝑡𝜔14 + 𝜔15 𝑡𝜔9 + 𝜔14⁄ > 4 4 6 6 9 9 𝑓4 ∪ {𝜔21} ℎ44 

I 4 5 8 9 9 10 𝑓4 ∪ {𝜔26} ℎ45 

𝑍2 =< 𝜔 𝑡⁄ > 4 4 5 5 6 6 𝑓4 ∪ {𝜔27} ℎ46 

𝑆3 =< 𝜔7 𝑡⁄ , 𝑡𝜔13 + 𝜔13 𝑡⁄ 𝜔6 + 𝜔7 > 5 5 5 5 5 5  𝑓5 ∪ {𝜔7} ℎ47 

𝑍2 =< 𝑡𝜔17 + 𝜔23 𝑡𝜔16 + 𝜔17⁄ > 5 5 6 6 10 10 𝑓5 ∪ {𝜔10} ℎ48 

𝑍2 =< 𝑡 + 𝜔6 𝑡 + 1⁄ > 5 5 7 7 9 9 𝑓5 ∪ {𝜔16} ℎ49 

𝑍2 =< 𝜔6(𝑡 + 1) 𝑡 + 𝜔6⁄ > 5 5 6 6 9 9 𝑓5 ∪ {𝜔21} ℎ50 

𝑆3 =< 𝜔11 𝑡⁄ , 𝑡𝜔 + 𝜔11 𝑡 + 1⁄ > 6 6 6 6 6 6 𝑓6 ∪ {𝜔11} ℎ51 

𝑍2 =< 𝜔(𝑡 + 1) 𝑡 + 𝜔⁄ > 6 6 7 7 9 9  𝑓6 ∪ {𝜔19} ℎ52 

𝑆3 =< 𝑡 + 𝜔8 𝑡 + 1⁄ , 𝑡𝜔24 + 𝜔17 𝑡⁄ > 9 9 9 9 9 9 𝑓9 ∪ {𝜔24} ℎ53 



 

IHJPAS. 2025,38(2) 

353 
 

Proof: To say two 5-sets 𝐴 = {∞, 0,1, 𝑎4, 𝑎5} and 𝐵 = {𝑏1, 𝑏2, 𝑏3, 𝑏4, 𝑏5} projectively 

equivalent we have to find a 2 × 2 matrix transform one of them to the other. So to find this 

matrix (if exists), we will construct a 2 × 2 matrix, say 𝑇, transform the three points ∞, 0,1 

in 𝐴 to order three points  

in 𝐵, say 𝑏1, 𝑏2, 𝑏3. Now if {𝑎4, 𝑎5}𝑇 = {𝑏4, 𝑏5}, then we say that 𝐴 and 𝐵 are projectively 

equivalent.  

Since each 4-set in Table 2 gives 8 orbits as in Table 3, so we have eight 5-sets from each 

4-set; that is, we have forty 5-sets.   

𝒰1 = 𝒯1⋃{𝜔2}, 𝒰2 = 𝒯1⋃{𝜔3}, 𝒰3 = 𝒯1⋃{𝜔4}, 𝒰4 = 𝒯1⋃{𝜔5}, 

𝒰5 = 𝒯1⋃{𝜔6}, 𝒰6 = 𝒯1⋃{𝜔10}, 𝒰7 = 𝒯1⋃{𝜔11}, 𝒰8 = 𝒯1⋃{𝜔16}, 

𝒰9 = 𝒯2⋃{𝜔5}, 𝒰10 = 𝒯2⋃{𝜔3}, 𝒰11 = 𝒯2⋃{𝜔4}, 𝒰12 = 𝒯2⋃{𝜔6}, 

𝒰13 = 𝒯2⋃{𝜔7}, 𝒰14 = 𝒯2⋃{𝜔8}, 𝒰15 = 𝒯2⋃{𝜔10}, 𝒰16 = 𝒯2⋃{𝜔12}, 

𝒰17 = 𝒯3⋃{𝜔}, 𝒰18 = 𝒯3⋃{𝜔2}, 𝒰19 = 𝒯3⋃{𝜔5}, 𝒰20 = 𝒯3⋃{𝜔6}, 

𝒰21 = 𝒯3⋃{𝜔7}, 𝒰22 = 𝒯3⋃{𝜔8}, 𝒰23 = 𝒯3⋃{𝜔9}, 𝒰24 = 𝒯3⋃{𝜔16}, 

𝒰25 = 𝒯4⋃{𝜔}, 𝒰26 = 𝒯4⋃{𝜔2}, 𝒰27 = 𝒯4⋃{𝜔3}, 𝒰28 = 𝒯4⋃{𝜔9}, 

𝒰29 = 𝒯4⋃{𝜔12}, 𝒰30 = 𝒯4⋃{𝜔13}, 𝒰31 = 𝒯4⋃{𝜔14}, 𝒰32 = 𝒯4⋃{𝜔19}, 

𝒰33 = 𝒯5⋃{𝜔}, 𝒰34 = 𝒯5⋃{𝜔2}, 𝒰35 = 𝒯5⋃{𝜔4}, 𝒰36 = 𝒯5⋃{𝜔9}, 

𝒰37 = 𝒯5⋃{𝜔11}, 𝒰38 = 𝒯5⋃{𝜔12}, 𝒰39 = 𝒯5⋃{𝜔14}, 𝒰40 = 𝒯5⋃{𝜔16}. 
𝑓1 = 𝒰1 projectively equivalents to 𝒰9; 

𝑓2 = 𝒰2 projectively equivalents to 𝒰10, 𝒰13, 𝒰26, 𝒰33; 

𝑓3 = 𝒰3 projectively equivalents to 𝒰16, 𝒰17, 𝒰37, 𝒰38; 

𝑓4 = 𝒰4 projectively equivalents to 𝒰15, 𝒰19, 𝒰29, 𝒰36; 

𝑓5 = 𝒰5 projectively equivalents to 𝒰12, 𝒰20, 𝒰23, 𝒰25; 

𝑓6 = 𝒰6 projectively equivalents to 𝒰7, 𝒰24, 𝒰30, 𝒰39; 

𝑓7 = 𝒰8 projectively equivalents to 𝒰31; 

𝑓8 = 𝒰11 projectively equivalents to 𝒰18; 

𝑓9 = 𝒰14 projectively equivalents to 𝒰21, 𝒰27, 𝒰28, 𝒰34; 

𝑓10 = 𝒰22 projectively equivalents to 𝒰35; 

To find the stabilizer group of each 5-set, the same technique explained above has been 

used with replacement of the set 𝐵 by 𝐴.    

The same procedure will be used in Theorems (6), (7), (8) and (10). 

3.3. The 6-set 

The 6-sets are constructed by adding one point from each orbit to the corresponding 5-sets. 

The projective group 𝐺𝑓𝑖
 splits 𝑓𝑖

𝑐 , 𝑖 = 1, … ,11 into a number of orbits. 

Theorem (6): In 𝑃𝐺(1,32), there are 53 projectively inequivalent 6-sets , summarized in the 

Table 5. 

3.4. The 7-sets until 17-sets 

The 7-sets are constructed by adding one point from each orbit to the corresponding 6-sets. 

The projective group 𝐺ℎ𝑖
 splits ℎ𝑖

𝑐, 𝑖 = 1, … ,53, into a number of orbits. 

Theorem 7: In 𝑃𝐺(1,32), there are 148 projectively inequivalent 7-sets , summarized in 

Table 6. 

 

 

 

 

 



 

IHJPAS. 2025,38(2) 

354 
 

Table 6. Inequivalent 7-sets 

SG Type of 6-set 7-set Symbol 

𝑍2 =< 𝜔4 𝑡⁄ > 1 1 2 2 8 31 31 ℎ1 ∪ {𝜔4} 𝑒1 

I 1 2 2 9 14 14 15 ℎ1 ∪ {𝜔5} 𝑒2 

I 1 2 3 10 22 27 35 ℎ1 ∪ {𝜔6} 𝑒3 

I 1 3 4 11 19 23 52 ℎ1 ∪ {𝜔7} 𝑒4 

I 1 4 5 12 15 18 22 ℎ1 ∪ {𝜔8} 𝑒5 

I 1 5 5 13 14 14 20 ℎ1 ∪ {𝜔9} 𝑒6 

I 1 4 5 6 14 25 46 ℎ1 ∪ {𝜔10} 𝑒7 

𝑍2 =< 𝜔2(𝑡 + 1) 𝑡 + 𝜔2⁄ > 1 1 6 6 15 15 24 ℎ1 ∪ {𝜔11} 𝑒8 

𝑍2 =< 𝑡𝜔20 + 𝜔23 𝑡𝜔19 + 𝜔20⁄ > 1 1 7 7 16 23 23 ℎ1 ∪ {𝜔12} 𝑒9 

I 1 3 4 7 14 22 50 ℎ1 ∪ {𝜔13} 𝑒10 

I 1 4 5 7 21 33 48 ℎ1 ∪ {𝜔14} 𝑒11 

I 1 6 7 18 20 34 51 ℎ1 ∪ {𝜔15} 𝑒12 

I 1 3 5 6 19 42 44 ℎ1 ∪ {𝜔16} 𝑒13 

𝑍2 =< 𝜔3 𝑡⁄ > 1 3 3 4 4 30 30 ℎ1 ∪ {𝜔17} 𝑒14 

𝑍2 =< 𝜔2(𝑡 + 1) 𝑡 + 𝜔2⁄ > 2 2 8 8 9 29 29 ℎ2 ∪ {𝜔5} 𝑒15 

I 2 3 9 10 15 20 30 ℎ2 ∪ {𝜔6} 𝑒16 

I 2 4 10 14 18 27 28 ℎ2 ∪ {𝜔7} 𝑒17 

I 2 5 11 21 24 36 45 ℎ2 ∪ {𝜔8} 𝑒18 

I 2 5 12 14 14 16 27 ℎ2 ∪ {𝜔9} 𝑒19 

I 2 4 6 13 31 42 45 ℎ2 ∪ {𝜔10} 𝑒20 

𝑍2 =< 𝑡 + 𝜔4 𝑡 + 1⁄ > 2 2 7 7 15 15 32 ℎ2 ∪ {𝜔12} 𝑒21 

I 2 4 10 16 33 37 45 ℎ2 ∪ {𝜔13} 𝑒22 

I 2 4 7 14 24 27 44 ℎ2 ∪ {𝜔14} 𝑒23 

I 2 5 6 9 14 30 34 ℎ2 ∪ {𝜔15} 𝑒24 

I 2 3 14 18 30 38 46 ℎ2 ∪ {𝜔16} 𝑒25 

I 2 3 19 23 29 30 40 ℎ2 ∪ {𝜔17} 𝑒26 

I 2 4 6 7 30 35 38 ℎ2 ∪ {𝜔18} 𝑒27 

I 2 5 7 14 18 30 36 ℎ2 ∪ {𝜔19} 𝑒28 

I 2 4 9 20 24 25 30 ℎ2 ∪ {𝜔20} 𝑒29 

I 2 5 7 14 21 27 33 ℎ2 ∪ {𝜔21} 𝑒30 

𝑍2 =< 𝜔4(𝑡 + 1) 𝑡 + 𝜔4⁄ > 2 2 6 6 23 23 36 ℎ2 ∪ {𝜔23} 𝑒31 

I 2 5 13 16 24 27 28 ℎ2 ∪ {𝜔24} 𝑒32 

I 2 5 21 25 37 41 45 ℎ2 ∪ {𝜔25} 𝑒33 

I 2 4 14 27 32 34 38 ℎ2 ∪ {𝜔26} 𝑒34 

I 2 3 21 22 34 43 45 ℎ2 ∪ {𝜔27} 𝑒35 

𝑍2 =< 𝑡 + 𝜔2 𝑡 + 1⁄ > 2 2 19 19 28 39 39 ℎ2 ∪ {𝜔28} 𝑒36 

𝑍2 =< 𝜔2 𝑡⁄ > 2 2 10 21 21 27 27 ℎ2 ∪ {𝜔29} 𝑒37 

I 3 4 14 22 25 40 47 ℎ3 ∪ {𝜔7} 𝑒38 

I 3 5 11 22 25 43 49 ℎ3 ∪ {𝜔8} 𝑒39 

I 3 5 13 19 25 42 50 ℎ3 ∪ {𝜔9} 𝑒40 

𝑍2 =< 𝑡𝜔19 + 𝜔29 𝑡𝜔17 + 𝜔19⁄ > 3 3 6 41 41 48 48 ℎ3 ∪ {𝜔10} 𝑒41 

𝑍2 =< 𝑡 + 𝜔12 𝑡 + 1⁄ > 3 3 7 31 31 37 37 ℎ3 ∪ {𝜔12} 𝑒42 

I 3 4 14 22 36 46 49 ℎ3 ∪ {𝜔13} 𝑒43 

I 3 4 7 15 34 35 46 ℎ3 ∪ {𝜔14} 𝑒44 

I 3 5 6 14 15 40 43 ℎ3 ∪ {𝜔15} 𝑒45 

𝑍2 =< 𝑡 + 𝜔2 𝑡 + 1⁄ > 3 3 42 42 47 49 49 ℎ3 ∪ {𝜔16} 𝑒46 

𝑍2 =< 𝜔2(𝑡 + 1) 𝑡 + 𝜔2⁄ > 3 3 33 33 38 43 43 ℎ3 ∪ {𝜔17} 𝑒47 

I 3 5 6 20 22 44 46 ℎ3 ∪ {𝜔18} 𝑒48 

I 3 3 6 7 14 14 52 ℎ3 ∪ {𝜔19} 𝑒49 

I 3 4 14 15 27 30 49 ℎ3 ∪ {𝜔20} 𝑒50 

I 3 5 14 15 38 49 50 ℎ3 ∪ {𝜔24} 𝑒51 

I 3 5 19 24 31 33 41 ℎ3 ∪ {𝜔25} 𝑒52 

I 3 4 4 14 15 35 51 ℎ3 ∪ {𝜔26} 𝑒53 

𝑍2 =< 𝜔2 𝑡⁄ > 3 3 45 45 48 52 52 ℎ3 ∪ {𝜔27} 𝑒54 

I 4 5 10 11 27 30 47 ℎ4 ∪ {𝜔8} 𝑒55 

I 4 5 11 14 16 38 40 ℎ4 ∪ {𝜔9} 𝑒56 

𝑍2 =< 𝑡 + 𝜔2 𝑡 + 1⁄ > 4 4 18 37 37 48 48 ℎ4 ∪ {𝜔13} 𝑒57 

I 4 4 6 14 21 33 50 ℎ4 ∪ {𝜔15} 𝑒58 

I 4 5 7 8 23 39 42 ℎ4 ∪ {𝜔19} 𝑒59 

I 4 5 8 14 19 22 42 ℎ4 ∪ {𝜔24} 𝑒60 

I 4 5 14 23 23 29 35 ℎ4 ∪ {𝜔25} 𝑒61 



 

IHJPAS. 2025,38(2) 

355 
 

SG Type of 6-set 7-set Symbol 

𝑍2 =< 𝜔2 𝑡⁄ > 4 4 20 31 31 41 41 ℎ4 ∪ {𝜔26} 𝑒62 

𝑍2 =< 𝑡 + 𝜔2 𝑡 + 1⁄ > 5 5 9 9 10 10 40 ℎ5 ∪ {𝜔9} 𝑒63 

𝑍2 =< 𝜔2(𝑡 + 1) 𝑡 + 𝜔2⁄ > 5 5 28 28 45 45 53 ℎ5 ∪ {𝜔24} 𝑒64 

𝑍2 =< 𝜔2 𝑡⁄ > 5 5 19 19 22 22 52 ℎ5 ∪ {𝜔25} 𝑒65 

𝑍2 =< 𝑡 + 𝜔2 𝑡 + 1⁄ > 6 6 7 7 29 31 31 ℎ6 ∪ {𝜔18} 𝑒66 

I 8 10 23 27 28 30 42 ℎ8 ∪ {𝜔6} 𝑒67 

𝑍2 =< 𝑡𝜔13 + 𝜔14 𝑡𝜔6 + 𝜔13⁄ > 8 8 11 27 27 39 39  ℎ8 ∪ {𝜔7} 𝑒68 

𝑍2 =< 𝜔8(𝑡 + 1) 𝑡 + 𝜔8⁄ > 8 8 12 19 19 21 21 ℎ8 ∪ {𝜔8} 𝑒69 

I 8 13 14 19 22 23 38 ℎ8 ∪ {𝜔9} 𝑒70 

I 8 14 14 29 31 31 37 ℎ8 ∪ {𝜔10} 𝑒71 

I 8 14 15 16 19 25 42 ℎ8 ∪ {𝜔11} 𝑒72 

I 8 16 24 29 32 36 39 ℎ8 ∪ {𝜔12} 𝑒73 

I 8 14 22 31 33 35 39 ℎ8 ∪ {𝜔13} 𝑒74 

I 8 18 21 23 24 31 34 ℎ8 ∪ {𝜔15} 𝑒75 

I 8 15 18 19 20 23 29 ℎ8 ∪ {𝜔16} 𝑒76 

I 9 10 11 12 13 25 28 ℎ9 ∪ {𝜔7} 𝑒77 

I 9 13 14 21 30 41 45 ℎ9 ∪ {𝜔9} 𝑒78 

I 9 10 14 24 27 30 38 ℎ9 ∪ {𝜔10} 𝑒79 

I 9 15 19 27 31 45 46 ℎ9 ∪ {𝜔11} 𝑒80 

I 9 16 21 27 28 36 45 ℎ9 ∪ {𝜔12} 𝑒81 

I 9 14 18 23 27 27 42 ℎ9 ∪ {𝜔15} 𝑒82 

𝑍2 =< 𝑡𝜔25 + 𝜔30 𝑡𝜔24 + 𝜔25⁄ > 9 9 19 19 28 23 43 ℎ9 ∪ {𝜔16} 𝑒83 

𝑍2 =< 𝑡𝜔14 + 𝜔15 𝑡𝜔9 + 𝜔14⁄ > 9 9 22 22 44 45 45 ℎ9 ∪ {𝜔21} 𝑒84 

I 10 14 14 41 45 45 48 ℎ10 ∪ {𝜔10} 𝑒85 

I 10 14 14 25 30 30 46 ℎ10 ∪ {𝜔13} 𝑒86 

𝑍2 =< 𝑡 + 𝜔6 𝑡 + 1⁄ > 10 10 19 19 45 45 49 ℎ10 ∪ {𝜔16} 𝑒87 

𝑍2 =< 𝜔6(𝑡 + 1) 𝑡 + 𝜔6⁄ > 10 10 22 22 28 28 50 ℎ10 ∪ {𝜔21} 𝑒88 

I 11 13 19 22 40 42 49 ℎ11 ∪ {𝜔9} 𝑒89 

I 11 14 20 23 42 46 50 ℎ11 ∪ {𝜔10} 𝑒90 

I 11 14 15 22 33 33 41 ℎ11 ∪ {𝜔11} 𝑒91 

I 11 14 27 33 37 45 43 ℎ11 ∪ {𝜔13} 𝑒92 

I 11 14 14 16 18 35 50 ℎ11 ∪ {𝜔15} 𝑒93 

I 11 14 21 28 30 44 45 ℎ11 ∪ {𝜔20} 𝑒94 

I 11 14 14 19 22 42 53 ℎ11 ∪ {𝜔21} 𝑒95 

I 12 14 19 20 22 23 24 ℎ12 ∪ {𝜔19} 𝑒96 

I 13 14 14 24 34 40 46 ℎ13 ∪ {𝜔10} 𝑒97 

I 13 14 15 20 22 43 52 ℎ13 ∪ {𝜔11} 𝑒98 

I 13 14 16 18 21 33 44 ℎ13 ∪ {𝜔12} 𝑒99 

I 13 15 19 23 36 43 44 ℎ13 ∪ {𝜔16} 𝑒100 

𝑍2 =< 𝑡 + 𝜔9 𝑡 + 1⁄ > 13 21 21 29 29 39 39  ℎ13 ∪ {𝜔20} 𝑒101 

I 13 14 27 33 36 45 53 ℎ13 ∪ {𝜔25} 𝑒102 

I 14 14 30 33 45 47 50 ℎ14 ∪ {𝜔13} 𝑒103 

I 14 14 18 21 23 31 32 ℎ14 ∪ {𝜔15} 𝑒104 

I 14 21 33 36 37 40 41 ℎ14 ∪ {𝜔20} 𝑒105 

I 14 15 21 22 23 31 48 ℎ14 ∪ {𝜔21} 𝑒106 

I 14 23 29 34 39 42 44 ℎ14 ∪ {𝜔22} 𝑒107 

I 14 15 18 22 24 25 35 ℎ14 ∪ {𝜔23} 𝑒108 

I 14 14 20 25 36 38 44 ℎ14 ∪ {𝜔24} 𝑒109 

I 14 14 19 19 39 43 49 ℎ14 ∪ {𝜔27} 𝑒110 

I 14 27 30 34 35 36 50 ℎ14 ∪ {𝜔28} 𝑒111 

I 14 23 28 30 33 42 45 ℎ14 ∪ {𝜔29} 𝑒112 

I 15 16 19 20 21 31 34 ℎ15 ∪ {𝜔12} 𝑒113 

I 14 14 15 15 34 44 52 ℎ15 ∪ {𝜔13} 𝑒114 

I 15 19 21 31 36 43 52 ℎ15 ∪ {𝜔16} 𝑒115 

I 15 21 22 33 37 48 49 ℎ15 ∪ {𝜔21} 𝑒116 

𝑍2 =< 𝑡𝜔14 + 𝜔25 𝑡𝜔23 + 𝜔14⁄ > 15 15 23 23 31 31 39 ℎ15 ∪ {𝜔22} 𝑒117 

I 14 15 19 24 29 42 43 ℎ15 ∪ {𝜔23} 𝑒118 

I 15 22 23 42 50 51 52 ℎ15 ∪ {𝜔26} 𝑒119 

I 15 22 27 28 30 49 52 ℎ15 ∪ {𝜔29} 𝑒120 

I 16 21 23 33 39 42 53 ℎ16 ∪ {𝜔20} 𝑒121 

I 14 16 19 22 23 49 52 ℎ16 ∪ {𝜔21} 𝑒122 

I 14 16 22 22 27 30 43 ℎ16 ∪ {𝜔25} 𝑒123 

I 14 21 29 31 41 42 46 ℎ17 ∪ {𝜔20} 𝑒124 



 

IHJPAS. 2025,38(2) 

356 
 

SG Type of 6-set 7-set Symbol 

I 14 14 21 21 24 37 38 ℎ17 ∪ {𝜔23} 𝑒125 

I 14 23 25 31 33 34 48 ℎ17 ∪ {𝜔24} 𝑒126 

I 14 14 36 39 40 42 42 ℎ17 ∪ {𝜔25} 𝑒127 

I 14 20 27 28 33 40 45 ℎ17 ∪ {𝜔29} 𝑒128 

I 18 19 22 31 33 39 52 ℎ18 ∪ {𝜔16} 𝑒129 

𝑍2 =< 𝜔(𝑡 + 1) 𝑡 + 𝜔⁄ > 19 19 25 27 27 29 29 ℎ19 ∪ {𝜔24} 𝑒130 

I 14 20 21 31 35 42 48 ℎ20 ∪ {𝜔20} 𝑒131 

I 21 25 27 38 40 41 45 ℎ21 ∪ {𝜔24} 𝑒132 

I 21 27 30 32 33 35 37 ℎ21 ∪ {𝜔28} 𝑒133 

I 14 23 31 33 44 46 51 ℎ23 ∪ {𝜔25} 𝑒134 

I 23 27 31 37 39 45 50 ℎ23 ∪ {𝜔28} 𝑒135 

I 29 30 31 38 39 42 45 ℎ29 ∪ {𝜔10} 𝑒136 

I 29 31 33 34 36 38 42 ℎ29 ∪ {𝜔13} 𝑒137 

𝑍2 =< 𝑡𝜔16 𝑡𝜔10 + 𝜔16⁄ > 30 30 31 31 48 48 51 ℎ30 ∪ {𝜔10} 𝑒138 

I 30 33 40 45 46 48 50 ℎ30 ∪ {𝜔13} 𝑒139 

I 30 31 35 40 42 44 45 ℎ30 ∪ {𝜔22} 𝑒140 

𝑍2 =< 𝑡𝜔29 𝑡𝜔25 + 𝜔29⁄ > 30 30 34 37 37 41 41 ℎ30 ∪ {𝜔25} 𝑒141 

I 

 

31 33 41 42 44 48 50 ℎ31 ∪ {𝜔13} 𝑒142 

𝑍2 =< 𝑡𝜔30 + 𝜔21 𝑡𝜔26 + 𝜔30⁄ > 33 33 35 49 49 52 52 ℎ33 ∪ {𝜔22} 𝑒143 

𝑍2 =< 𝑡𝜔30 + 𝜔24 𝑡𝜔29 + 𝜔30⁄ > 33 33 37 42 42 45 45 ℎ33 ∪ {𝜔25} 𝑒144 

I 35 37 38 41 46 47 48 ℎ35 ∪ {𝜔25} 𝑒145 

I 40 43 44 49 50 52 53 ℎ40 ∪ {𝜔16} 𝑒146 

𝑍2 =< 𝑡 + 𝜔 𝑡 + 1⁄ > 41 43 43 45 45 49 49  ℎ41 ∪ {𝜔16} 𝑒147 

𝑍2 =< 𝑡𝜔9 + 𝜔24 𝑡𝜔24 + 𝜔9⁄ > 42 42 43 43 46 52 52 ℎ42 ∪ {𝜔16} 𝑒148 

 

By using the same technique, the inequivalent 𝑘-sets, 𝑘 = 8,9,10,11 have been found with their 

stabilizer groups. 

Theorem 8: In 𝑃𝐺(1,32)   

(i) There are 481 inequivalent 8-sets and their stabilizer given in Table 7. 

(ii) There are 1240 inequivalent 9-sets and their stabilizer given in Table 8. 

(iii) There are 2964 inequivalent 10-sets and their stabilizer given in Table 9. 

(iv) There are 6049 inequivalent 11-sets and their stabilizer given in Table 10. 

 

Table 7. Inequivalent 8-set 

SG. NO. 

I 371 

𝑍2 105 

𝑍2 × 𝑍2 × 𝑍2 5 

 

Table 8. Inequivalent 9-set 

SG. NO. 

I 1125 

𝑍2 100 

𝑍3 5 

𝑆3 5 

𝑍2 × 𝑍2 × 𝑍2 5 

 

Table 9. Inequivalent 10-set 

No. SG. 

2691 I 

273 𝑍2 

 

 



 

IHJPAS. 2025,38(2) 

357 
 

Table 10. Inequivalent 11-set 

No. SG. 

5776 𝐼 

272 𝑍2 

1 𝐷11 

 

Example 9: The 8-set, 9-set and 11-set with large stabilizer group are  

(i) There are five 8-sets with stabilizer group of type 𝑍2 × 𝑍2 × 𝑍2 

{∞, 0,1, 𝜔, 𝜔2, 𝜔3, 𝜔15, 𝜔19}, 𝑍2 × 𝑍2 × 𝑍2 =<
𝜔3

𝑡
,

𝑡+𝜔3

𝑡+1
,

𝜔16𝑡+𝜔17

𝜔14𝑡+𝜔16 >. 

(ii) There are five 9-sets with stabilizer group of type  𝑍2 × 𝑍2 × 𝑍2 

{∞, 0,1, 𝜔, 𝜔2, 𝜔3, 𝜔4, 𝜔12, 𝜔23},  𝑍2 × 𝑍2 × 𝑍2 =<
𝜔4

𝑡
,

𝑡+𝜔4

𝑡+1
,

𝜔11𝑡+𝜔12

𝜔8𝑡+𝜔11 >. 

 (iii) There is a unique 11-set with stabilizer group of type 𝐷11 

{∞, 0,1, 𝜔, 𝜔2, 𝜔3, 𝜔4, 𝜔7, 𝜔13, 𝜔22, 𝜔28} ,  𝐷11 =< 𝑡 + 𝜔7,
𝜔10

𝜔6𝑡+𝜔13 >. 

Since the numbers of 𝑘-sets, 𝑘 = 12, … ,17 are very large, so we consider only sets that have 

non-trivial different stabilizers. 

Theorem 10: In 𝑃𝐺(1,32), there are more than 

(i) 493 inequivalent 12-sets. 

(ii) 5077 inequivalent 13-sets.  

(iii) 2583 inequivalent 14-sets.  

(iv) 288 inequivalent 15-sets. 

(v) 2412 inequivalent 16-sets.  

(vi) 697 inequivalent 17-sets. 

The stabilizer groups of 𝑘-sets, 𝑘 = 12, … ,17 given in Table 11. 

 

Example 11: The 12-set,13-set, 15-set and 17-set with large stabilizer groups as follows  

(i) There are ten 12-sets with stabilizer group of type 𝑆3 

{∞, 0,1, 𝜔, 𝜔2, 𝜔3, 𝜔4, 𝜔6, 𝜔8, 𝜔9, 𝜔11, 𝜔16}, 𝑆3 =<
𝜔6𝑡+𝜔14

𝜔5𝑡+𝜔6
,

𝜔22

𝜔10𝑡+𝜔16
>. 

(ii) There are 35 13-sets with stabilizer group of type 𝑉4 

{∞, 0,1, 𝜔, 𝜔2, 𝜔3, 𝜔4, 𝜔5, 𝜔10, 𝜔12, 𝜔24, 𝜔26, 𝜔18}, 𝑉4 =<
𝜔5

𝑡
,

𝑡+𝜔5

𝑡+1
>. 

(iii) There are four 15-sets with stabilizer group of type 𝑆3 

{∞, 0,1, 𝜔, 𝜔2, 𝜔3, 𝜔4, 𝜔5, 𝜔6, 𝜔8, 𝜔9, 𝜔20, 𝜔11, 𝜔24, 𝜔16}, 𝑆3 =<
𝜔14𝑡+𝜔23

𝜔6𝑡+𝜔14
,

𝜔6𝑡+𝜔12

𝑡
>. 

(iv) There is a unique 17-set with stabilizer group of type  𝑍2 × 𝑍2 × 𝑍2 × 𝑍2   

{∞, 0,1, 𝜔, 𝜔2, 𝜔3, 𝜔4, 𝜔5, 𝜔6, 𝜔7, 𝜔14, 𝜔24, 𝜔8, 𝜔15, 𝜔17, 𝜔22, 𝜔25},  

𝑍2 × 𝑍2 × 𝑍2 × 𝑍2 =<
𝜔8

𝑡
,

𝑡+𝜔8

𝑡+1
,

𝜔11𝑡+𝜔18

𝜔10𝑡+𝜔11
,

𝜔9𝑡+𝜔15

𝜔7𝑡+𝜔9
>. 

Table 11. The stabilizer group 

𝒌-sets SG 

12-set 438: 𝑍2    10:𝑍3    35:𝑉4    10:𝑆3 

13-set 4240:𝐼      448: 𝑍2    35: 𝑉4     

14-set 2583:𝐼 

15-set 282: 𝑍2    2: 𝑍3    4: 𝑆3     

16-set 2397:𝐼     15:𝑍2   

17-set 667: 𝑍2    29: 𝑉4    1: 𝑍2 × 𝑍2 × 𝑍2 × 𝑍2    



 

IHJPAS. 2025,38(2) 

358 
 

3.5. Partition of 𝑃𝐺(1,32)  

(i) The projective line 𝑃𝐺(1,32), can be partitioned depending on each projectively distinct 

11-set into three 11-sets for example: 

The complement of 11-set 𝒦11 = 𝑒1⋃{𝜔5, 𝜔6, 𝜔7, 𝜔8}, which has stabilizer group 𝑍2, is  

𝒦11
𝐶 = {𝜔9, 𝜔10, … , 𝜔30}. 𝒦11

𝐶  can be partitioned into two 11-sets as follows: 

Let Μ1 = {𝜔9, 𝜔10, 𝜔11, 𝜔12, 𝜔14, 𝜔17, 𝜔20, 𝜔21, 𝜔22, 𝜔25, 𝜔26}, and Μ1 = {𝜔13, 𝜔15, 

𝜔16, 𝜔18, 𝜔19, 𝜔23, 𝜔24, 𝜔27, 𝜔28, 𝜔29, 𝜔30}. 𝒦11
𝐶  is projectively equivalent to Μ1 by the 

matrix [ [𝜔23, 𝜔3 ], [ 𝜔8, 𝜔14] ]. Then the triple {𝒦11, Μ1; Μ2} formed a partition of  

𝑃𝐺(1,32)  by 11-sets.   

(ii) The projective line 𝑃𝐺(1,32), can be partitioned depending on the five projectively 

distinct 4-sets into eight 4-sets plus say {∞} point for example: 

(a) Partition by eight 4-sets of type 𝑁1:  

There are 8184 of 4-sets of type 𝑁1.  

{ 0, 1, 𝜔, 𝜔2}, 𝜆 = 𝜔17 ;  { 𝜔3, 𝜔4, 𝜔5, 𝜔13}, 𝜆 = 𝜔30 ;  { 𝜔6, 𝜔7, 𝜔8, 𝜔16}, 𝜆 =

𝜔30 ;  {𝜔9, 𝜔10, 𝜔11, 𝜔19}, 𝜆 = 𝜔30 ;  {𝜔12, 𝜔14, 𝜔15, 𝜔20}, 𝜆 =

𝜔18 ;   {𝜔17, 𝜔18, 𝜔21, 𝜔23}, 𝜆 = 𝜔18 ;   {𝜔22, 𝜔25, 𝜔27, 𝜔28}, 𝜆 = 𝜔30 ; 

{ 𝜔24, 𝜔26, 𝜔29, 𝜔30}, 𝜆 = 𝜔18 . 

(b) Partition by eight 4-sets of type 𝑁2:  

There are 8184 of 4-sets of type 𝑁2.  

{ 0, 1, 𝜔, 𝜔7}, 𝜆 = 𝜔29 ;  { 𝜔2, 𝜔3, 𝜔4, 𝜔13}, 𝜆 = 𝜔3 ;  { 𝜔5, 𝜔6, 𝜔8, 𝜔12}, 

 𝜆 = 𝜔29 ;  {𝜔9, 𝜔10, 𝜔11, 𝜔14}, 𝜆 = 𝜔26 ;  {𝜔15, 𝜔16, 𝜔17, 𝜔20}, 𝜆 =

𝜔26 ;   {𝜔18, 𝜔19, 𝜔21, 𝜔25}, 𝜆 = 𝜔29 ;   {𝜔22, 𝜔26, 𝜔27, 𝜔30}, 𝜆 = 𝜔5 ; { 𝜔23, 𝜔24, 𝜔28,

𝜔29}, 𝜆 = 𝜔29. 

(c) Partition by eight 4-sets of type 𝑁3:  

There are 8184 of 4-sets of type 𝑁3.  

{ 0, 1, 𝜔, 𝜔6}, 𝜆 = 𝜔4 ;  { 𝜔2, 𝜔3, 𝜔4, 𝜔5}, 𝜆 = 𝜔25 ;  { 𝜔7, 𝜔8, 𝜔9, 𝜔10}, 𝜆 =

𝜔25 ;  {𝜔11, 𝜔12, 𝜔13, 𝜔14}, 𝜆 = 𝜔25 ;   {𝜔15, 𝜔16, 𝜔17, 𝜔18}, 𝜆 =

𝜔25 ;   {𝜔19, 𝜔20, 𝜔21, 𝜔22}, 𝜆 = 𝜔25 ;  {𝜔23, 𝜔24, 𝜔25, 𝜔27}, 𝜆 =

𝜔6;  {𝜔26, 𝜔28, 𝜔29, 𝜔30}, 𝜆 = 𝜔25.  

(d) Partition by eight 4-sets of type 𝑁4:  

There are 8184 of 4-sets of type 𝑁4.  

{ 0, 1, 𝜔, 𝜔3}, 𝜆 = 𝜔9 ;  { 𝜔2, 𝜔4, 𝜔5, 𝜔7}, 𝜆 = 𝜔7 ;  { 𝜔6, 𝜔8, 𝜔9, 𝜔11}, 𝜆 =

𝜔7 ;  {𝜔10, 𝜔12, 𝜔13, 𝜔15}, 𝜆 = 𝜔7 ;   {𝜔14, 𝜔16, 𝜔17, 𝜔19}, 𝜆 =

𝜔7 ;   {𝜔18, 𝜔20, 𝜔21, 𝜔23}, 𝜆 = 𝜔7 ;  {𝜔22, 𝜔24, 𝜔26, 𝜔29}, 𝜆 =

𝜔16;  {𝜔25, 𝜔27, 𝜔28, 𝜔30}, 𝜆 = 𝜔7. 

(e) Partition by eight 4-sets of type 𝑁5:  

There are 8184 of 4-sets of type 𝑁5.  

{ 0, 1, 𝜔, 𝜔4}, 𝜆 = 𝜔20 ;  { 𝜔2, 𝜔3, 𝜔5, 𝜔6}, 𝜆 = 𝜔12 ;  { 𝜔7, 𝜔8, 𝜔9, 𝜔14}, 𝜆 =

𝜔23 ;  {𝜔10, 𝜔11, 𝜔12, 𝜔17}, 𝜆 = 𝜔23 ;   {𝜔13, 𝜔15, 𝜔16, 𝜔24}, 𝜆 =

𝜔8 ;   {𝜔18, 𝜔19, 𝜔20, 𝜔25}, 𝜆 = 𝜔23 ;  {𝜔21, 𝜔22, 𝜔23, 𝜔28}, 𝜆 =

𝜔23;  {𝜔26, 𝜔27, 𝜔29, 𝜔30}, 𝜆 = 𝜔12. 

(f) Partition by four 4-sets of type 𝑁5 and:  

There are 8184 of 4-sets of type 𝑁5.  

Type 𝑁1: { 0, 1, 𝜔, 𝜔2}, 𝜆 = 𝜔17 ;  

Type 𝑁2: { 𝜔3, 𝜔4, 𝜔5, 𝜔8}, 𝜆 = 𝜔26 ; 

Type 𝑁3: { 𝜔6, 𝜔7, 𝜔9, 𝜔20}, 𝜆 = 𝜔25 ; 



 

IHJPAS. 2025,38(2) 

359 
 

Type 𝑁4: {𝜔10, 𝜔11, 𝜔12, 𝜔16}, 𝜆 = 𝜔24 ;  

Type 𝑁5: {𝜔13, 𝜔14, 𝜔15, 𝜔21}, 𝜆 = 𝜔20 ;  

Type 𝑁5: {𝜔17, 𝜔18, 𝜔19, 𝜔24}, 𝜆 = 𝜔23 ;  

Type 𝑁5: {𝜔22, 𝜔26, 𝜔28, 𝜔30}, 𝜆 = 𝜔12;  

Type 𝑁5: {𝜔23, 𝜔25, 𝜔27, 𝜔29}, 𝜆 = 𝜔19. 

 

4. Conclusions 

      In this paper, we introduce and proved there are 5, 11, 53, 148, 481, 1240, 2963, 6049, 

493, 5077, 2583, 288, 2412, 697, projectively inequivalent 𝑘-sets, 𝑘 = 4, … ,17, 

respectively. The Kind of stabilizer groups which appeared were 𝐼, 𝑍2, 𝑍3, 𝑉4, 𝑆3, 𝑍2 ×

𝑍2 × 𝑍2, 𝐷11.  Order of the projective line 𝑃𝐺(1,32), which is 33, divisible by 3 and 11 

only. So we are able to partition the line into three distinct 11-sets. Also, if we exclude the 

point ∞ from 𝑃𝐺(1,32) we are able to partition the line into eight 4-sets of type 

𝑁1, 𝑁2, 𝑁3, 𝑁4, 𝑁5.   

 

Acknowledgment 

     The authors would like to thank the University of Mustansiriyah, College of Science, and 

Department of Mathematics for their motivation and support. 

 

Conflict of Interest  

     The authors declare that they have no conflicts of interest. 

 

Funding 

     There is no funding for the article. 

 

References 

1. Hirschfeld JWP. Projective Geometries Over Finite Fields. 2nd ed. Oxford Mathematical 

Monographs. The Clarendon Press, Oxford University Press, New York; 1998. ISBN: 0-19-

850295-8. https://doi.org/10.1093/oso/9780198502951.001.0001 

2. Sadeh AR. The Classification of k-Arcs and Cubic Surfaces with Twenty-Seven Lines Over the 

Field of Eleven Elements [Doctoral dissertation]. University of Sussex: Brighton, UK; 1984. 

3. Ali AH. Classification of Arcs in the Galois Plane of Order Thirteen [Doctoral dissertation]. 

University of Sussex: Brighton, UK; 1993. 

4. Al-Zangana EB, Ibrahm MM. Classification of Subsets in Finite Projective Line Over Galois Field 

of Order Twenty-Seven. J Phys Conf Ser. 2021;1818. https://doi.org/10.1088/1742-

6596/1818/1/012087 

5. Hirschfeld JWP, Al-Seraji NA. The Geometry of Line of Order Seventeen and its Application to 

Error-Correcting Code. Al-Mustansiriyah J Sci. 2013;24(5):217-230. 

https://doi.org/10.13140/RG.2.2.32059.98082 

6. Al-Seraji NA, Musa FA. Classification of Subsets in Finite Projective Line. J Phys Conf Ser. 

2021;1879. https://doi.org/10.1088/1742-6596/1879/3/032028 

7. Al-Seraji NA, Essa AK. Classification of the Projective Line Over Galois Field of Order 31. Iraqi 

J Sci. 2023;64(4):1881-1901. https://doi.org/10.24996/ijs.2023.64.4.28 

8. Radhi JSh, Al-Zangana EB. Extension of Cap by Size and Degree in the Space PG(3,11). Ibn Al-

Haitham J Pure Appl Sci. 2023;36(2):375–382. https://doi.org/10.30526/36.2.3025 

9. Younis AA, Kasm Yahya NY. The Construction (k+1; n)-Arcs and (k+2; n)-Arcs from Incomplete 

(k;n)-Arc in PG(3,q). Palestine J Math. 2023;12(Special Issue I):107–129.  

https://doi.org/10.1093/oso/9780198502951.001.0001
https://doi.org/10.1088/1742-6596/1818/1/012087
https://doi.org/10.1088/1742-6596/1818/1/012087
https://doi.org/10.13140/RG.2.2.32059.98082
https://doi.org/10.1088/1742-6596/1879/3/032028
https://doi.org/10.24996/ijs.2023.64.4.28
https://doi.org/10.30526/36.2.3025


 

IHJPAS. 2025,38(2) 

360 
 

10. Sulaimaan AE, Kasm Yahya NY. The Reverse Construction of Complete (k;n)-Arcs in Three-

Dimensional Projective Space PG(3,4). J Phys Conf Ser. 2020;1591. https://doi.org/10.1088/1742-

6596/1591/1/012078 

11. Kasm Yahya NY. Applications Geometry of Space in PG(3,P). J Interdiscip Math. 

2022;25(2):285–297. https://doi.org/10.1080/09720502.2021.1885818 

12. Hamada N, Maruta T, Oya Y. A Necessary and Sufficient Condition for the Existence of an (n;r)-

Arc in PG(2, q) and its applications. Serdica J Comput. 2012;6:253-266. 

https://doi.org/10.55630/sjc.2012.6.253-266 

13. Kareem FF. A Complete (k,r)-Cap in PG(3,p) Over Galois Field GF(4). Ibn Al-Haitham J Pure 

Appl Sci. 2011;24(2):236-247. https://jih.uobaghdad.edu.iq/index.php/j/article/view/743 

14. Kareem FF, Kadhum SJ. A (k,l)-Span in Three-Dimensional Projective Space PG(3,p) over Galois 

Field where p=4. Ibn Al-Haitham J Pure Appl Sci. 2013;19(80):659-672. 

https://doi.org/10.35950/cbej.v19i80.7999 

15. Kareem FF. The Construction of Complete (k;n)-arcs in 3-Dimensional Projective Space Over 

Galois Field GF(4). J Coll Educ Al-Mustansiriyah Univ. 2013;1:183-196. 

https://www.researchgate.net/publication/280626270 

16. Hamed ZS, Hirschfeld JWP. A complete (48,4)-Arc in the Projective Plane Over the Field of Order 

Seventeen. Baghdad Sci J. 2021;18(4):1238. https://doi.org/10.21123/bsj.2021.18.4.1238 

17. Günay G, Lavrauw M. On pencils of Cubics on the Projective Line over Finite Fields of 

Characteristic >3. math.CO, math.AG. 2021. https://doi.org/10.48550/arXiv.2104.04756 

18. Han D, Ren Y. The Maximal Length of q-ary MDS Elliptic Codes Is Close to q/2. Int Math Res 

Not. 2023;rnad271. https://doi.org/10.1093/imrn/rnad271 

19. The GAP Group. GAP Reference Manual. (accessed on 30 October 2022). 

20. Thomas AD, Wood GV. Group Tables. Shiva Mathematics Series; 2. Shiva Publishing Ltd: UK; 

1980. ISBN: 978-0906812020 

 

 

 

https://doi.org/10.1088/1742-6596/1591/1/012078
https://doi.org/10.1088/1742-6596/1591/1/012078
https://doi.org/10.1080/09720502.2021.1885818
https://doi.org/10.55630/sjc.2012.6.253-266
https://jih.uobaghdad.edu.iq/index.php/j/article/view/743
https://doi.org/10.35950/cbej.v19i80.7999
https://www.researchgate.net/publication/280626270
https://doi.org/10.21123/bsj.2021.18.4.1238
https://doi.org/10.48550/arXiv.2104.04756
https://doi.org/10.1093/imrn/rnad271