Simple affine racks

Let $A$ be an abelian group and $T \in$ Aut$(A)$, then A is a rack with $x \triangleright y = (1 − T )x + T y$. This is called an affine rack and denoted by $Q_{A,T}$.

 The simple affine racks are given by the racks $Q_{\mathbb{F}_{p}^{t} ,T}= (\mathbb{F}_{p}^{t} , T )$, where $p$ is a prime number and $T$ is the companion matrix of a monic irreducible polynomial  $f \in \mathbb{F}_{p}[X]$ of degree $t$, different from $X$ and $X - 1$, see [AG], Racks.

Alternatively, set $q = p^t$ and identify $\mathbb{F}_q$ with $\mathbb{F}_p^{t}$. Then the action of $T$ corresponds to multiplication by $a$, which is the class of $X$ in $\mathbb{F}_p [X]/(f)$. Note that $a$ generates $\mathbb{F}_q$ over $\mathbb{F}_p$.

Problem [AFGaV1, Question 6].
Find the proper subracks of $Q_{\mathbb{F}_{q},a}$.

One expects that the simple affine racks have very few subracks. In fact, they have no abelian subracks with more than one element [AFGV1, Remark 3.13]. In the following we list some partial results supporting this idea.

Proposition. [AFGaV1, Proposition 4.1]
If $a$ generates $\mathbb{F}_q^{\times}$, then any proper subrack of $Q_{\mathbb{F}_q ,a}$ is trivial.

Proposition. [AFGaV1, Proposition 4.2].
Let $1\neq a \in \mathbb{F}_p^{\times}$. Then any proper subrack of the affine rack $Q_{\mathbb{F}_p ,a}$ is trivial.

We now list several examples of pairs of affine racks and cocycles $(X, q)$ such that $\dim \mathfrak{B}(X, q) < \infty$; we give the dimension, the top degree and the reference where the example appeared.

$X$	rk	q	Relations	$\dim \mathfrak{B}(X, q)$	top	Ref.
$Q_{\mathbb{Z}/5,2}$	5	-1	10 in degree 2 1 in degree 4	$1280=5.4^4$	$16=4^2$	[AG]
$Q_{\mathbb{Z}/5,3}$	5	-1	10 in degree 2 1 in degree 4	$1280=5.4^4$	$16=4^2$	dual of the preceding

$Q_{\mathbb{Z}/7,3}$
7
-1
21 in degree 2
1 in degree 6
$326592=7.6^6$
$36=6^2$
[G2]
$Q_{\mathbb{Z}/7,5}$
7
-1
21 in degree 2
1 in degree 6
$326592=7.6^6$
$36=6^2$
dual of the
preceding