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$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What is the smallest group not known to be a Galois group over $\mathbb{Q}$?

Variants have been asked here before (e.g. Which small finite simple groups are not yet known to be Galois groups over Q? specifically for simple groups), but I explicitly don't want to restrict to simple groups, since this let's one get to rather lofty orders all while dodging the true difficulty of solving embedding problems. Of course I'm aware that results in the literature are usually not custom-made to answer these kind of "trivia" questions, so here are some thoughts.

My own guess is that the answer may be as small as $\SL_2(13)$ (which has order $2184$), so that I'm essentially asking two types of sub-questions:

  1. Is there any quotable reference or reasonably short argument for the claim that all groups of order smaller than $2184$ have been realized? (Or alternatively, am I in fact missing some smaller open case?)

Since a counterexample would have to be (non-simple) nonsolvable involving a very small nonsolvable composition factor (concretely, $\PSL_2(q)$ with $q\le 11$), I guess I might be able to piece this together from various results in the literature about solvability of central embedding problems, split embedding problems, and embedding problems with (certain) nonabelian simple kernel, although I haven't checked in full and the full list of relevant groups is not that small after all. (A short argument might be of the form "every embedding problem is either of this or that type, and these types are known to be solvable when only the couple of smallest simple groups are involved.")

Update: After writing some code filtering the relevant nonsolvable groups, using some known general results such as solvability of split embedding problems with nilpotent kernel (Shafarevich), realizability of all central extensions of $A_n$ and $S_n$ and solvability of all embedding problems with kernel $A_n$ ($n\ne 6$) (see Malle-Matzat's book for the latter two), the orders I'm actually struggling to rule out directly are $16\cdot 60$, $32\cdot 60$, $8\cdot 168$, $4\cdot 360$ and $6\cdot 360$ (where in each case I've factored by the order of the relevant nonabelian composition factor), although in particular the last two cases might fall with a bit more effort.

  1. Is $\SL_2(13)$ indeed an open case? (And if not, what is the smallest open $\SL_2(p)$?)

The reason why $\SL_2(p)$ looks difficult is that, in order to solve the embedding problem from $\PSL_2(p)$, one necessarily needs to begin with a (suitable) totally real $\PSL_2(p)$-extension, which doesn't go too well with the "classical" realizations à la Shih from rigidity theory (although $p=7$ and $11$ have been solved). I'm not too familiar with the more recent realizations of $\PSL_2(p)$ (for all primes $p$) by Zywina, although I would have thought they don't yield totally real extensions either. In short, is there either a recent reference (say, from the last decade) stating that $p=13$ (or similar) is open, or otherwise does someone know a realization? (Just to make sure: the standard databases seem to know no such realization.)

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    $\begingroup$ It is true that there are $49487367289$ groups of order $1024$, and surely no one has looked at all of them one at a time, but presumably there are general theorems about groups of order $p^n$ that deal with those groups all at once. If there are such theorems, and also theorems for other shapes of factorizations, it could well be that $2184=2^3\times3\times7\times13$ is the smallest unsettled case. But I, too, defer to the experts. $\endgroup$ Commented Mar 21, 2023 at 22:32
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    $\begingroup$ @GerryMyerson Solvable groups are all fine due to Shafarevich's theorem. Also all simple groups in this range are known to occur, hence my mentioning that the potential counterexample must be nonsimple nonsolvable. This leaves only the proper multiples of 60, 168, 360, 504 and 660 - some of which are also fine rather directly, e.g. since direct products are no problem and $PGL_2(11)$ and $SL_2(11)$ have been realized, one can check that the $660$ can actually be dropped again. The most tedious ones might be groups containing a composition factor $A_5$. $\endgroup$ Commented Mar 21, 2023 at 23:07
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    $\begingroup$ For question 2, I think it's still open, e.g. see the talk [the absolute Galois group of the rational numbers numbers] given by Ravi Ramakrishna, youtube.com/watch?v=Lb4VfjgrwAw&ab_channel=JaredONGARO (play from 27:10). Also, the blog galoisrepresentations.com/2013/04/24/inverse-galois-problem claimed that it's open for $p>13$ in 2013. $\endgroup$
    – stupid boy
    Commented Mar 22, 2023 at 11:55
  • $\begingroup$ @stupidboy That's very helpful, thanks! $\endgroup$ Commented Mar 22, 2023 at 14:16

1 Answer 1

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Doing an internet search I found the paper Groups of small order as Galois groups over $\mathbb{Q}$ by Jack Sonn, from 1989. Theorem 1 of that paper asserts that every group of order less than 672 is a Galois group for an extension of $\mathbb{Q}$. The nontrivial central extensions of ${\rm PGL}(2,7)$ are listed as the (then) current undecided cases of order 672.

The Math Review of that paper suggests that the proof was incomplete, but patchable. The paper has no references listed in MathSciNet, so it is possible that this is still the state of the art. (You might consider contacting the author of the paper, or of the math review, to see what they say about the current situation.)

There are four papers that reference Sonn's paper listed in "Google Scholar". As far as I could ascertain, none of those four asserted a larger number.

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    $\begingroup$ These groups are no longer open, as observed by Malle in Section 12 of mathematik.uni-kl.de/~malle/download/pargal.pdf , the reason essentially being the existence of a local-global principle for central embedding problems (together with the availability of sufficiently nice $PGL_2(7)$ extensions). Which (together with all other pre-existing evidence) might make the author of that paper the right person to ask. $\endgroup$ Commented Mar 22, 2023 at 1:10
  • $\begingroup$ For one's information, it seems that the groups you are talking about are $\operatorname{SmallGroup}(672,1044)$ and $\operatorname{SmallGroup}(672,1045)$ (from the link Groups of order 672). $\endgroup$ Commented Aug 24, 2023 at 8:46

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