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I am interested in understanding (at least roughly, if no such a description exists) the group of automorphisms for the group $UT(n,p)$, of unitriangular matrices over the field $\mathbb{F}_p$ on $p$-elements. Unfortunately the online searching I've carried out wasn't quite helpful. If no description exists, maybe some information how the automorphisms behave on the set of subgroups of it may be helpful too.

For instance when $n=3$, I know we have that any $\phi \in Aut(UT(3,p))$, permutes the pairs of non-commuting elements of $UT(3,p)$ (which in that case happens always to be a generating set) and the subgroups of order $p^2$.

So am wondering if something similar exists (of course modified somehow) in the general case too.

If the above sounds quite general, the case where $n=4$, is of great importance for me either. Any comment might be useful, and of course references too.

Thanks!

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  • $\begingroup$ I haven't given this any thought at all, but computer calculations with $3 \le n \le 8$ strongly suggest that the order of the automorphism group of ${\rm UT}(n,p)$ is $p^3(p-1)^2(p+1)$ when $n=3$, and $2p^e(p-1)^{n-1}$ with $e = (n^2+n-4)/2$, when $n > 3$. So, for $n=4$, this would give $2p^8(p-1)^3$. $\endgroup$ – Derek Holt Sep 11 '17 at 12:04
  • $\begingroup$ Derek thank you very much for your reply. For $n=3$ I know that this is the case indeed, since I can write down explicitly what's going on. For $n=4$ though the situation becomes really hard and what you wrote is helpful indeed. What I am trying to understand is mainly how the automoprhisms permute the subgroups of order $p^3$ (for $n=4$ at least) and because I don't have access to any computational system at the moment is difficult to do any computation by hand. $\endgroup$ – mayer_vietoris Sep 11 '17 at 12:11
  • $\begingroup$ I did some calculations on the orbits of the automorphism group of ${\rm UT}(4,p)$ on subgroups of order $p^3$ for $p=3,5,7$. At this stage, computations are starting to take longer, but I could go a bit further if necessary. In summary, for $p=3,5,7$, there are respectively $109$, $391$, and $953$ conjugacy classes of subgroups of order $p^3$, and the automorphism group has respectively $19$, $21$ and $23$ orbits on these subgroups. $\endgroup$ – Derek Holt Sep 11 '17 at 12:56
  • $\begingroup$ Thank you very much for all the effort you have put on this. One question only, when you mean "...and the automorphism group has respectively 19, 21 and 23 orbits on these subgroups.", you mean that the $Aut(UT(4,p))$-action on the set of subgroups of order $p^3$ yields these 19,21 and 23 orbits respectively or something different? Because it confuses me a little bit. $\endgroup$ – mayer_vietoris Sep 11 '17 at 13:30
  • $\begingroup$ Yes that is what I meant. I did the calculation in Magma by computing orbits of the outer automorphism group on the conjugacy classes of subgroups of order $p^3$. I can provide more detailed information on this necessary. $\endgroup$ – Derek Holt Sep 11 '17 at 13:35
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  • for size = 3 https://groupprops.subwiki.org/wiki/Unitriangular_matrix_group:UT(3,p)#Automorphisms Automorphisms The automorphisms essentially permute the subgroups of order
    containing the center, while leaving the center itself unmoved.

  • over $F_2$, any size: Maginnis J. S., (1993/11)."Outer Automorphisms of Upper Triangular Matrices." Journal of Algebra 161(2): 267-270. Abstract: The outer automorphism group of the upper triangular matrices over the field of two elements is calculated. A. J. Weir (Proc. Amer. Math. Soc.6 (1955), 454-464) performed a similar calculation for Fields of odd characteristic, and we borrow the term extremal auto ....

  • Any field(may be even ring) International Journal of Algebra, Vol. 7, 2013, no. 15, 723 - 733 HIKARI Ltd, The Automorphism Group of the Group of Unitriangular Matrices over a Field1 Ayan Mahalanobis Abstract. This paper finds a set of generators for the automorphism group of the group of unitriangular matrices over a field. Most of this paper is an exposition of the work of V.M. Lev˘chuk, part of which is in Russian. Some proofs are of my own. From the paper: The automorphism group of the group of unitriangular matrices over a field was studied by many [2–4]. In this direction, the first paper was in Russian, published by Pavlov in 1953. Pavlov studies the automorphism group of unitriangular matrices over a finite field of odd prime order. Weir [4] describes the automorphism group of the group of unitriangular matrices over a finite field of odd characteristic. Maginnis 3 describes it for the field of two elements and finally Lev˘chuk 2 describes the automorphism group of the group of unitriangular matrices over an arbitrary ring. In this expository article, we shall study the automorphism group of the group of unitriangular matrices over an arbitrary field F. There are two most commonly used non-abelian finite p-groups in the literature. One is the group 1Research supported by a NBHM research grant. 724 Ayan Mahalanobis of unitriangular matrices over a field and the other is the extra-special pgroups. Weir [4] and Lev˘chuk 2 worked on the automorphism group of the unitriangular matrices.

PS I remember I have seen some papers with quite explicit description of automorphism, and I thought I collected such links in comments to my old MO-question: Representation theory of p-groups in particular upper tringular matrices over F_p, but it cannot find it now again, hm-m... I'll try to find more...

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  • $\begingroup$ Definitely very helpful answer (of course +1). However I will read it through thoroughly (were I am is really late now) tomorrow, and if any question arises will let you know. Cheers! $\endgroup$ – mayer_vietoris Sep 11 '17 at 22:22
  • $\begingroup$ @DerekHolt thank you very much too, I tagged you because you wrote that you haven't thought this through again, and the above by a quick view seems interesting, hence might be helpful for you too. $\endgroup$ – mayer_vietoris Sep 11 '17 at 22:24

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