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Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite field of order $p$. Let $U_{n}$ denote the unitriangular group of $n\times n$ upper triangular matrices with ones on the diagonal, over $\mathbb{F}_{p}$. Are there groups such as their automorphism group is $U_{n}$?.

Any example or reference will be helpful. Thank you all.

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    $\begingroup$ @Nick Gill: I thought that Gaschutz proved that ${\rm Out}(P)$ has order divisible by $p$ for any finite $p$-group $P$. $\endgroup$ Mar 30, 2020 at 16:26
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    $\begingroup$ @StevenStadnicki It has $p^{n(n-1)/2}$ elements, so yes. $\endgroup$
    – Wojowu
    Mar 30, 2020 at 16:28
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    $\begingroup$ @StevenStadnicki :Yes, $U_{n}$ is always a $p$-group. If $p^{k}$ is the smallest power of $p$ greater than or equal to $n$, we have $u^{p^{k}} = I$ for each $u \in U_{n}.$ $\endgroup$ Mar 30, 2020 at 16:30
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    $\begingroup$ @NickGill: It seems that groups of order $p$ are the only exceptions to Gaschutz's theorem. $\endgroup$ Mar 30, 2020 at 17:41
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    $\begingroup$ $U_3(p=2)\cong D_8\cong Aut(D_8)$. $\endgroup$ Mar 30, 2020 at 21:22

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