Use Algebra of Group Action to Find Special Types of Caps in 𝑷𝑮(𝟑, 𝟏𝟑)

Group action on the projective space 𝑃𝐺(3, 𝑞) is a method which can be used to construct some geometric objects for example cap. We constructed new caps in 𝑃𝐺(3,13) of degrees 2, 3, 4, 7,14 and sizes 2 , 4 , 5 , 7 , 10 , 14 , 17 , 20 , 28 , 34 , 35 , 68 , 70 , 85 , 119 , 140 , 170 , 238 , 340 , 476 , 595 , 1190 . Then the incomplete caps are extended to complete caps

The aim of this paper is to use the algebra of group action to find the special type of caps (3,13). This idea has been used to calculate caps in (3,8), see [7], (3,11), see [8], and for (3,23), see [9]. Regarding to (3,13), in [10] the authors studied this space, the case of space's partition by span.

Basic definitions and preliminaries Definition 1:[2]
A ( , )-cap in ( ≥ 3, ) is a set of points such that no + 1 points are collinear, but at most points of which lie in any line. Here is called degree of the ( , )-cap. The number of all points of index will be denoted by .
The sequence ( 0 , … , ) will represent the secant distribution and the sequences ( 0 , … , ), ≤ ⌊ ⌋ (⌊ ⌋denotes the smallest integer less than or equal to ) refer to the index distribution.

Definition 4:[11]
Let be a group and a non-empty set. An action of on is a map × ⟶ denoted ( , ) ↦ such that 1 = and (ℎ ) = ( ℎ) for all in and , ℎ in . For an element of , the orbit of or the orbit through is the subset : = { | ∈ } of . These orbits partitioned the set into disjoint subsets; that is, = ⋃ ∈ .

Lagrange's Theorem 5:[11]
If is a finite group and is a subgroup of , then | | divides | |.
As it is well-known in a finite group, the number of elements of each orbit divides the order of finite group . Also, the order of each element of divides the order of .
The companion matrix is an element of (4, ), and it is a cyclic projectivity; that is, it has an order equals to the order of (3, ), say 3 ( ). Therefore, for each positive integer divided 3 ( ), the set 〈 〉 will be a subgroup of a group 〈 〉 with order such that = 3 ( ).  • Depending on these integers , the subgroups 〈 〉 of the group will be computed from the group 〈 〉. • The GAP (Groups-Algorithms-programing) programming [12] is used to execute the design algorithms to find the elements of the subgroups 〈 〉, to find orbits, secant distribution and index distribution for each cap.

Main results
In this section, we introduce new caps formed from the acts of cyclic subgroups of the group 〈 〉 on (3,13) . Theorem: There are 22 equivalence classes in (3,13) formed caps with the following details: 1. The orbits from the action of 〈 2 〉 on (3,13) are incompletes of size 1190 and will be caps of degree 14.

The orbits from the action of 〈 4 〉 on
(3,13) are completes of size 595 and will be caps of degree 7.

The orbits from the action of 〈 5 〉 on
(3,13) are incompletes of size 476 and will be caps of degree 14. 4. The orbits from the action of 〈 7 〉 on (3,13) are completes of size 340 and will be caps of degree 4. 5. The orbits from the action of 〈 10 〉 on (3,13) are incompletes of size 238 and will be caps of degree 14. 6. The orbits from the action of 〈 14 〉 on (3,13) are completes of size 170 and will be caps of degree 2. 7. The orbits from the action of 〈 17 〉 on (3,13) are incompletes of size 140 and will be caps of degree 14. 8. The orbits from the action of 〈 20 〉 on (3,13) are incompletes of size 119 and will be caps of degree 7. 9. The orbits from the action of 〈 28 〉 on (3,13) are incompletes of size 85 and will be caps of degree 2. 10. The orbits from the action of 〈 34 〉 on (3,13) are incompletes of size 70 and will be caps of degree 14. 11. The orbits from the action of 〈 35 〉 on (3,13) are incompletes of size 68 and will be caps of degree 3. 12. The orbits from the action of 〈 68 〉 on (3,13) are incompletes of size 35 and will be caps of degree 7. 13. The orbits from the action of 〈 70 〉 on (3,13) are incompletes of size 34 and will be caps of degree 2. 14. The orbits from the action of 〈 85 〉 on (3,13) are incompletes of size 28 and will be caps of degree 14. 15. The orbits from the action of 〈 119 〉 on (3,13) are incompletes of size 20 and will be caps of degree 2. 16. The orbits from the action of 〈 140 〉 on (3,13) are incompletes of size 17 and will be caps of degree 2.  (3,13) are incompletes of size 10 and will be caps of degree 2. 19. The orbits from the action of 〈 340 〉 on (3,13) are incompletes of size 7 and will be caps of degree 7. 20. The orbits from the action of 〈 476 〉 on (3,13) are incompletes of size 5 and will be caps of degree 2. 21. The orbits from the action of 〈 595 〉 on (3,13) are incompletes of size 4 and will be caps of degree 2. 22. The orbits from the action of 〈 1190 〉 on (3,13) are incompletes of size 2 and will be caps of degree 2.
This group will act on the space (3,13), and then partitioned the space into orbits each of them has points and projectively equivalents by . Thus in our work, we will take the first orbit, say . Below, the details of the orbits related to the 22 divisors of 2380. 1. For, = 2 the orbit 2 = {1 + 2 | = 0, … ,1189} has 1190 points, and the numbers of point's intersection between 2 and lines of (3,13) are as follows: There are 85 lines with no intersection-points, 2380 lines with four intersection-points, 5950 lines with five intersection-points, 4760 lines with six intersection-points, 4760 lines with seven intersection-points, 4760 lines with eight intersection-points, 5950 lines with nine intersectionpoints, 2380 lines with ten intersection-points, and 85 lines with fourteen intersection-points, so 2 is (1190,14)-cap.
Since the degree of 2 is 14, which is the number of lines-points, then 0 is 3 (13) − | 2 | = 2380 − 1190 = 1190; that is, 2 is incomplete. Also, the large complete cap formed from 2 of degree 14 is the whole space.  There are 6120 lines with no intersection-points, 2040 lines with one intersection-points, 14790 lines with two intersection-points, 2040 lines with three intersection-points, and 6120 lines with four intersection-points, so 7 is (340,4)-cap.