Subgroups and Orbits by Companion Matrix in Three Dimensional Projective Space

: The aim of this paper is to construct cyclic subgroups of the projective general linear group over 𝐹 23 from the companion matrix, and then form caps of various degrees in 𝑃𝐺(3,23) . Geometric properties of these caps as secant distributions and index distributions are given and determined if they are complete. Also, partitioned of 𝑃𝐺(3,23) into disjoint lines is discussed.

As special cases, the number of points (hyperplanes, by duality), lines and planes in ( , ) are as follows: Definition 1: ( ; )-cap, of degree is a set of points in ( ≥ 3, ) such that no + 1 points are collinear, but some of them are collinear. The set is called a complete ( ; )-cap if is maximal with respect to set-theoretical inclusion 1 . The maximum size of a cap of degree will be denoted by ( , ). The size of the smallest complete cap of degree will be denoted by ( , ). Definition 2: et be a cap of degree . An -secant of a in ( , ) is a line such that | ⋂ | = . The number of -secants of will be denoted by 1 . Let be a point not on the ( ; )-cap, . The number of -secants of through will be denoted by ( ). The number ( ) of -secants is called the index of with respect to . The set of all points of index will be denoted by and the cardinality of denoted by . The sequence ( 0 , … , ) will refer to secant distribution and the sequence ( 0 , … , ), ≤ ⌊ ⌋ to the index distribution. Definition 3: The group of projectivities of ( , ) is called the projective general linear group, ( + 1, ) 1 .
The elements of ( + 1, ) are non-singular matrices of dimension + 1, and its cardinality is The companion matrix has the property that it is cyclic of order ( , ) so, all points (hyperplanes) of ( , ) can be formed using the formula: ( ) = 0 , where 0 = [1, 0, . . . , 0] and from 0 to ( , ) − 1. Also, formed a cyclic group of ( , ) which is called the Singer group. According to Lagrange's theorem (group theory), any natural number divided the order of the Singer group generated by will give a cyclic subgroup of order . Therefore, the points of ( , ) will be partitioned into a number of classes from the act of Singer group on ( , ); that is, for a fixed natural number divided ( , ), the point related to point iff there is a natural number such that • ≤ ( , ) and = .
To know more about the three-dimensional projective space, which is of interest in this research, see source 3 .
The paper aims to study the geometric structure of classes formed from the Singer group of (4,23) like cap, secant distributions and index distributions. The topic of caps has been studied extensively by many researchers in finite projective spaces. For three dimensional projective space, some of them attempted to find the smallest complete caps or the largest complete caps as in 4,5,6 . A spectrum size of complete caps has been given for certain fields 7,8 . As for studying the caps in the three dimensional projective space, there are many researchers who have studied this concept 9 , and the link between it and the concepts of linear coding 10 and with cubic curves 11 . The idea of action of groups on the points of projective space to construct geometrical objects that appear in 12 for line and with respect to projective plane in 13,14 . In (3,23), (3, 23) = 12720 = ϕ(3, 23), and (3, 23) = 293090. The number 12720 has 2, 2, 2, 2, 3, 5 as prime factor integers so, there are 38 non-trivial divisors of (3,23 (3,23). The order of the companion matrix is also (2, 23) which is give cyclic subgroup, 〈 〉 of (4, 23) such that (3,23) invariant with respect to it. So, all elements of divided the order of and give cyclic subgroups of 〈 〉. Let denote these subgroups by , in . Any other cyclic subgroup of (4, 23) of order divided (3, 23) will be a copy isomorphic to for some in . Theorem 7: There are unique 38 equivalence classes up to projectivity of (3,23) of order in such that · = (3, 23). Proof. The action of the groups , ∈ in Lemma 6, on the projective space (3, 23), will divide the space points into equivalence classes (orbits) of order ∈ ; that is, = (3,23) . All these equivalence classes will be projectively equivalent by . Let denote the classes in the Theorem 7, by . The intersection of these classes has been tested with planes to find out the degrees of the caps that it formed. Also, its non-zero -secant distributions and 0 -distributions have computed.

Outline of Algorithm:
The algorithm that has been used to construct the caps by the action of the subgroups on the projective space (3,23) is summarized as below, and it has been executed by GAP programming 15 .
Step 1: For fixed in , construct the subgroup .
Step 2: Construct the orbit . Step 3: Compute the number of intersection points of with all lines of (3,23).
Step 4: The maximum point of intersection will determined as a degree of the cap .
Step 5: Compute the number of points out of that are not on the 3-secants to determine if it is complete or not. The class 53 is the union of the ten lines ℓ 2+53 , = 0, … ,9.

xii.
The class 530 is just the line ℓ 2 .
Remark 10: All compete caps in Theorem 8 have no points in common with lines in the 1-space cycle of length 530.

Conclusion:
From the study of group action in (3,23), new caps of different sizes and degrees have been found, and show that the maximum size of cap of degree 2 has many copies; that is, it is not unique. Also, it has been shown that the six complete caps in Corollary 9, have no point-contacts with lines in the 1-space cycle.