# Transitive Deficiency One Parallelisms of PG(3, 7)

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Construction

#### 2.1. Preliminaries

#### 2.2. Automorphism Groups

#### 2.3. A Parallelism Invariant under ${G}_{49}$

#### 2.4. The Isomorphism of Solutions

## 3. Methods

#### 3.1. Preliminaries

#### 3.2. Method 1

`S1Construct(2).`

void S1Construct(int Orb) { for(int i=1; i<=400; i++) { if(NotPossibleLineOrb(Orb, i)) continue; Put(Orb, i); if(Orb==14) WriteSpread(); else S1Construct(Orb+1); Take(Orb); } }

`NotPossibleLineOrb`returns true if some of the lines of the considered i-th line orbit under ${G}_{{7}_{1}}$ intersect lines of the already chosen

`Orb-1`orbits, or if no more orbits of this length can be added (there must be 8 orbits of length 1 and 6 orbits of length 7—Figure 1).

`Put`adds the orbit to the spread and

`Take`removes it. If the spread is ready, it is saved by

`WriteSpread`, and if more orbits have to be added,

`S1Construct`

`(Orb+1)`is called to choose the next orbit.

`PConstr(2)`, where

`allS=|L|`and

`SpreadOK`is called to check if the i-th spread possibility does not have common lines with the obtained until this moment partial parallelism.

void PConstr(int Spr) { for(int i=1; i<=allS; i++) { if(SpreadOK(Spr, i)) { PutSpread(Spr, i); if(Spr==8) WriteParallelism(); else PConstr(Spr+1); TakeSpread(Spr); } } }

#### 3.3. Method 2

`MakePar(2, 1).`

void MakePar(int Line, int Spr) { int Point = FirstMissingPoint(Line, Spr); for(int i = FirstLine[Point]; i<=LastLine[Point]; i++) { if(Possible(Line, i, Spr)) { PutLine(Line, i, Spr); if(Line==50) { if(RestrictionsOK(Spr) { if(Spr==8) WriteParallelism(); else MakePar(2, Spr+1); } } else MakePar(Line+1, Spr); TakeLine(Line, Spr); } } }

`FirstMissingPoint`returns the number of the first missing point in spread

`Spr`. Each point must be in one line of the spread. Therefore, we try to add only lines containing the first missing point. Their numbers are between

`FirstLine[Point]`and

`LastLine[Point]`.

`Possible`returns true if the considered line i has no common points with the lines of the current partial spread. If this is the case,

`PutLine`adds it to the current solution. We continue to add lines until all points are covered (i.e., the number of lines is 50) and when this happens,

`RestrictionsOK`checks if the imposed restrictions hold. If so,

`MakePar(2, Spr+1)`starts adding the next spread, or

`WriteParallelism`saves a ready parallelism.

`RestrictionsOK`solutions for ${S}_{1}$ for which there exists a line ${l}_{e}\notin {S}_{R}$ such that no element of H maps any line of ${O}_{{S}_{1}}$ to ${l}_{e}$ (because in this case transitivity on ${S}_{1}$ is not possible). We also save a list L of the different spreads to which ${S}_{1}$ is mapped by the elements of H. From the solutions for ${S}_{i},i=2,3,\dots ,8$ we reject by

`RestrictionsOK`those which are not in L and those which are in L, but for which there exists a line ${l}_{f}$ not contained in the partial solution and not contained in any of the spreads from L that can be added after ${S}_{i}$.

## 4. Results

#### 4.1. The Transitive Deficiency One Parallelisms of $\mathrm{PG}(3,7)$

**All**is the number of parallelisms with a full automorphism group of order

**Auts**. The rows below the double line show how many of these parallelisms are invariant under elation groups of given orders, namely

**E A**is the number of parallelisms invariant under an elation group of order

**A**.

#### 4.2. Uniform Deficiency One Parallelisms of $\mathrm{PG}(3,7)$

**All**is the number of parallelisms with a full automorphism group of order

**Auts**. The rows below the double line show how many of these parallelisms have a given number of spread orbits under their full automorphism groups, namely

**k orbits**is the number of parallelisms whose spreads are in

**k**orbits under the full automorphism group of the parallelism.

## 5. Discussion on the Obtained Results

- We obtain three parallelisms of Johnson type and it is shown in [32] that for $q=p$, an odd prime, the number of non-isomorphic Johnson-type parallelisms is exactly $(p-1)/2$.
- The order of the full automorphism groups of the parallelisms of Johnson type that we construct is either 4704, or 9408 and it is shown in [32] that the order should be two or four times the order of the full central collineation group of the regular spread.
- None of the transitive deficiency one parallelisms of $\mathrm{PG}(3,7)$ is self-dual, as shown in [28].

- There are 46 transitive deficiency one parallelisms of $\mathrm{PG}(3,7)$ which are invariant under an elation group of order 49 and have a regular deficiency spread. The three parallelisms from Johnson’s infinite class are among them.
- All the spreads of the constructed transitive deficiency one parallelism of $\mathrm{PG}(3,7)$ are Hall spreads and the deficiency spread is regular by assumption.
- The dual of a transitive deficiency one parallelism of $\mathrm{PG}(3,7)$ is a transitive deficiency one parallelism with the same order of the full automorphism group as the original parallelism, but it is not isomorphic to the original parallelism.
- The duals of the Johnson type parallelisms of $\mathrm{PG}(3,7)$ are not of the Johnson-type.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$GF\left(q\right)$ | - | the finite field with q elements with addition and |

multiplication defined on them | ||

$\mathrm{PG}(n,q)$ | - | the n-dimensional projective space over the finite field $GF\left(q\right)$ |

$P\Gamma L(m,q)$ | - | the projective general semi-linear group |

elation subgroup | - | a group of automorphisms of $\mathrm{PG}(3,q)$ which fixes all points of one line |

central collineation | - | elation (automorphism of $\mathrm{PG}(3,q)$ which fixes all points of one line) |

G | - | the group of automorphisms of the considered projective space $\mathrm{PG}(3,7)$, |

$G\cong P\Gamma L(4,7)$ ($G\cong P\Gamma L(n+1,q)$) | ||

${G}_{49}$ | - | an elation subgroup of G of order 49 |

${G}_{{7}_{1}},{G}_{{7}_{2}},\dots ,{G}_{{7}_{8}}$ | - | the eight subgroups of order 7 of ${G}_{49}$ |

${S}_{R}$ | - | the regular deficiency spread of the constructed parallelisms |

$N\left({G}_{49}\right)$ | - | the normalizer of ${G}_{49}$ in G, $N\left({G}_{49}\right)=\left\{g\in G|g{G}_{49}{g}^{-1}={G}_{49}\right\}$ |

N | - | the subgroup of $N\left({G}_{49}\right)$ which preserves ${S}_{R}$ |

H | - | $H=N/{G}_{49}$ |

${S}_{1},{S}_{2},\dots ,{S}_{8}$ | - | representatives (orbit leaders) of the spread orbits of length 7 |

${O}_{{S}_{1}},{O}_{{S}_{2}},\dots ,{O}_{{S}_{8}}$ | - | the spread orbits of length 7 of a parallelism invariant under ${G}_{49}$ |

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Group | Point Orbits | Line Orbits | ||||
---|---|---|---|---|---|---|

fixed | length 7 | length 49 | fixed | length 7 | length 49 | |

${G}_{{7}_{i}}$ | 8 | 56 | 0 | 57 | 343 + 56 | 0 |

${G}_{49}$ | 8 | 0 | 8 | 1 | 0 + 64 | 49 |

**Table 2.**Transitive deficiency one parallelisms of $\mathrm{PG}(3,7)$-order of the full automorphism groups and their elation subgroups.

Auts | 2352 | 4702 | 9408 | All |

All | 24 | 18 | 4 | 46 |

E 294 | 12 | 8 | – | 20 |

E 588 | 6 | 5 | 2 | 13 |

E 1176 | 6 | 3 | 1 | 10 |

E 2352 | – | 2 | 1 | 3 |

**Table 3.**Spread orbits and order of the full automorphism groups of the constructed uniform deficiency one parallelisms.

Auts | 49 | 294 | 588 | 1176 | 2352 | 4704 | 9408 | All |

All | 23,040 | 29,500 | 2258 | 168 | 34 | 18 | 4 | 55,022 |

9 orbits | 23,040 | 29,500 | 2 | – | – | – | – | 52,542 |

6 orbits | – | – | 1096 | 4 | – | – | – | 1100 |

5 orbits | – | – | 1160 | 8 | – | – | – | 1168 |

4 orbits | – | – | – | 88 | 2 | – | – | 90 |

3 orbits | – | – | – | 68 | 8 | – | – | 76 |

2 orbits | – | – | – | – | 24 | 18 | 4 | 46 |

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Topalova, S.; Zhelezova, S.
Transitive Deficiency One Parallelisms of PG(3, 7). *Mathematics* **2023**, *11*, 2458.
https://doi.org/10.3390/math11112458

**AMA Style**

Topalova S, Zhelezova S.
Transitive Deficiency One Parallelisms of PG(3, 7). *Mathematics*. 2023; 11(11):2458.
https://doi.org/10.3390/math11112458

**Chicago/Turabian Style**

Topalova, Svetlana, and Stela Zhelezova.
2023. "Transitive Deficiency One Parallelisms of PG(3, 7)" *Mathematics* 11, no. 11: 2458.
https://doi.org/10.3390/math11112458