subgroups of $\mathrm{GL}(n,q)$ of order $q^n-1$ are called Singer cycles; it is well-known that any two Singer cycles $S_1=\langle g_1\rangle$ and $S_2=\langle g_2\rangle$ are conjugate in $\mathrm{GL}(n,q)$.

In fact, let $p_{g_1}(x)$ be the minimal polynomial of $g_1$ over $\mathbb{F}_q$, and $\gamma$ be a primitive element of ${\mathbb{F}_{q^n}}$ with minimal polynomial $p_{g_1}(x)$ over ${{\mathbb F}_{q}}$. The set $\overline{S}_1=S_1\cup\{\mathbf{0}\}$ is an ${{\mathbb F}_{q}}$-subalgebra of ${{\mathbb F}_{q}}^{n\times n}$, isomorphic to ${\mathbb{F}_{q^n}}$ by the ${{\mathbb F}_{q}}$-linear map $\varphi$ mapping $(1,g_1,\ldots,g_1^{n-1})$ to $(1,\gamma,\ldots,\gamma^{n-1})$. Also, $\overline{S}_1$ is a field of order $q^n$ and $\varphi$ is a field ${{\mathbb F}_{q}}$-isomorphism. The same holds for $\overline{S}_2=S_2\cup\{\mathbf{0}\}$, so that there exists a field ${{\mathbb F}_{q}}$-isomorphism $\psi:\overline{S}_1\to\overline{S}_2$. Therefore, there exists $\hat{\psi}\in\mathrm{GL}(n,q)$ which conjugates $S_1$ to $S_2$. See also [@Huppert pag. 187] and [@Hiss Section 1.2.5 and Example 1.12].

Also the converse of Theorem \[th:MRDiff\] holds, in the sense that any MRD code as in the claim of that theorem is equivalent to $\mathcal{C}_{U,G}$ for some $U$ as in the assumption of Theorem \[th:MRDiff\].

\[th:MRDconverse\] Let $\mathcal{C}$ be an ${{\mathbb F}_{q}}$-linear MRD code with parameters $(t,n,q;n-\iota)$ such that $t\geq n$ and $|R(\mathcal{C})|=q^n$, contained in ${\rm Hom}({\mathbb{F}_{q^n}},W)$ with $W=V(t,q)$. Let $r=\dim_{R(\mathcal{C})}(\mathcal{C})$. Then the following holds.

-   $\iota+1$ divides $rn$ and $t=\frac{rn}{\iota+1}$.

-   