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Let $\mathbf{F}_q$ be finite field of order $q$, where $q$ is an odd-prime (or power of an odd prime). Is there an isomorphism between the following subgroups of unipotent $3\times 3$ matrices over $\mathbf{F}_q$:

$$\left\{ \begin{pmatrix}1 & x& y\\0&1&0\\0&0&1\end{pmatrix} \mid x,y\in \mathbf{F}_q\right\}, \text{ and } \left\{ \begin{pmatrix}1 & x& y\\0&1&x\\0&0&1\end{pmatrix} \mid x,y\in \mathbf{F}_q\right\}?$$

One way is to map generators to generators, and extend it to products. Is there any other isomorphism possible between the above mentioned subgroups?

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    $\begingroup$ They are both elementary groups of the same order. $\endgroup$
    – Derek Holt
    May 25, 2020 at 7:39

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Once the groups are isomorphic, the set of isomorphisms is a principal homogeneous space for the automorphism group (acting by composition on one of the sides).

So here the set of isomorphisms is in bijection with $\mathrm{GL}_{2e}(\mathbb{F}_p)$ where $q=p^e$ (since the group is the additive group of a $2e$-dimensional $\mathbb{F}_p$-vectorspace).

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    $\begingroup$ You're right, and there are even more automorphisms. The group is actually $C_p^{2e}$ if $q=p^e$. $\endgroup$
    – Lior Silberman
    May 25, 2020 at 7:42

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