Parallelisms of PG(3,4) invariant under an elementary abelian group of order 4

Abstract

This paper is a contribution to the classification of parallelisms in three-dimensional projective spaces over small finite fields of order q by computer. The smallest space in which parallelisms have not yet been classified is for $q=4.$ Partial results are available. The parallelisms admitting a nontrivial automorphism of odd prime order are known. Moreover, much is known about the case of parallelisms of ${{\mathrm{PG}}}(3,4)$ whose automorphism group is a two group. Namely, everything is known for two of the three possible groups of order two, as well as for cyclic groups of order 4. The present paper will settle the case of parallelisms whose automorphism group is elementary abelian of order 4. This leaves open the cases of parallelisms whose full automorphism groups are either trivial or a specific group of order two.

Appendices

A Proof of Lemma 3 using Orbiter

A computer calculation shows that $C_G(c_1)$ has 29 orbits on the elements of class 2C, one of which consists of the element $c_1$. Of the 28 orbits other than $c_1$, we reject 21 because $c_3=c_1c_2$ does not belong to class 2C. This leaves 7 possibilities for $c_2$ under the action of $C_G(c_1)$, summarized in Table 11.

Table 11 Orbits of $C_G(c_1)$ on 2C, with orbit representative $c_2$ such that $c_3 = c_1c_2\in 2C$

But these 7 cases do not correspond to the conjugacy classes of subgroups, because $c_1$ is distinguished. They are pairs of groups (H, K) with $H \le K$ with $H = \langle c_1 \rangle$ and all nontrivial elements of K belong to class 2C. We now have to consider all other choices for subgroups of K which could play the role of H. To do so, we let each of $c_2$ and $c_3$ play the role of $c_1$. We consider only one case, the other case is similar. Suppose we pick $c_2$ to play the role of $c_1$. To this end, we determine an element $g \in G$ such that $c_2^g=c_1,$ so $\langle c_2^g \rangle = H.$ We then check to which line in Table 11$c_1^g$ belongs. There are two cases. If $c_1^g$ belongs to the same orbits as $c_2$ under $C_G(c_1)$ we can determine an element $h\in C_G( c_1)$ such that $c_1^{gh} = c_2.$ In this case we have

$$\begin{aligned} K^{gh} = \{e, c_1,c_2,c_3\}^{gh} = \{e, c_1^{gh}, c_2^{gh}, c_3^{gh}\} = \{e, c_2,c_1,c_3^{gh}\} = \{e, c_1,c_2,c_1c_2\}=K, \end{aligned}$$

and hence $gh \in N_G(K).$ If $c_1^g$ belongs to a different case in Table 11, we eliminate that case. We then repeat the procedure for $c_3$, again finding $g\in G$ and possibly $h\in C_G( c_1)$ to either find another generator for $N_G(K)$ or to eliminate another row of Table 11. The set of gh for which we compute when we stay in the same case in Table 11 form a system of coset representatives for $N_G(\langle c_1,c_2,c_3\rangle )$ with respect to the subgroup that is listed as the stabilizer of $c_2$ in $C_G(c_1)$. After small calculations in the computer algebra system Orbiter [6], this yields the 5 groups listed in Table 2. Note that this procedure gives explicit generators for the normalizer $N_G(\langle c_1,c_2,c_3\rangle ).$

B Orbiter code for the construction of parallelisms

Orbiter is written in C++ and uses the unix command line. For a description, see the user’s guide at [7]. Makefiles or shell scripts can be used to perform simple programming of orbiter command, for instance using string variables. Makefiles allow to store multiple commands in one file, which is not the case with shell scripts. We will show some example code (makefile code) that was used to verify the claims made in the paper. Line numbers are shown for convenience (they are not part of the file). Makefiles are very sensitive to whitespace characters. For this reason, we show the code in a slightly enhanced form. Tabulator symbols are indicated as small triangles pointing to the right. The command below sets up makefile variables for the group, its normalizer, and for the spreads in ${\mathrm{PG}}(3,4)$. We define the group H1 as the first group from Lemma 3, and the group N1 to be the normalizer (of order 768). Both groups are then collected in the variable GROUP_AND_NORMALIZER:

The next step is to create a table of all spreads in ${\mathrm{PG}}(3,4).$ The following command does just that.

This command starts by defining the field $F={{\mathbb {F}}}_4$, the projective space $P={\mathrm{PG}}(3,4)$, and the table of spreads T. This command takes some computing time (about an hour on a conventional laptop). The spread tables are stored in files in a subdirectory SPREAD_TABLES_4. The parameter 2 refers to the vector space dimension of the spread elements (we are considering line-spreads). The parameter "0,1,2" refers to the three isomorphism types of spreads in ${\mathrm{PG}}(3,4)$ as in Lemma 2. The following command creates the orbits of H1 on spreads and writes a latex report:

The next command computes the 5-cliques on the graph of fix-spreads, and classifies them under the action of N1. As it turns out, there are 8 orbits.

The following command processes each of the orbits on fix points and computes the cliques on the graph of long orbits:

The final steps involve assembly of the packings from the long and short orbits, as well as isomorphism testing. For reasons of space, we omit these steps. The complete set of makefiles associated with the work described in this paper is available through the Orbiter distribution [6] (in the directory examples/BTZ2022).