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The icosahedral subgroup of $ G=PSU_2 \cong SO_3(\mathbb{R}) $ is a finite simple group which is also maximal closed in $ G $. Do other Lie groups admit maximal closed subgroups which are also finite and simple?

EDIT: as you can see from the comments below the original question was an attempt to ask something with a similar spirit but ultimately was misguided because I didn't understand Zariski density. The question as it is now is asked in the same spirit, "large" finite simple subgroups of Lie groups, but no longer makes any mention of zariski topology.

Ian Gershon Teixeira
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  • Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – Shaun Jan 12 '22 at 17:19
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    Hint: Finite subgroups are never Zariski dense. – Moishe Kohan Jan 12 '22 at 17:34
  • @MoisheKohan Are compact subgroups in general never Zariski dense? Or just finite subgroups? – Ian Gershon Teixeira Jan 12 '22 at 20:57
  • @IanGershonTeixeira: Never over real numbers, but they can be easily dense over complex numbers, e.g. $SU(n)$ in $SL(n,C)$. As for finite subgroups, they are never Zariski dense (in an algebraic Lie group of positive dimension) over real or complex numbers. Do you understand why? – Moishe Kohan Jan 12 '22 at 21:04
  • If Zariski topology was T_1 then points are closed so finite sets are always closed but I think Zariski is classic example of topology that is not T_1 right? – Ian Gershon Teixeira Jan 12 '22 at 22:55
  • @IanGershonTeixeira: I suggest that you read the definition of Zariski topology, say, on ${\mathbb C}^n$ and then try to prove the correct statement. – Moishe Kohan Jan 13 '22 at 01:45
  • Ok my bad. The Zariski topology on a variety is always $ T_1 $ since any point $ (a_1, \dots, a_n) $ is $ V(x_1-a_1, \dots, x_n-a_n) $. And Zariski topology on a variety is defined by taking basic closed sets to be simultaneous vanishing set of a collection of polynomials. So points are closed. Everything with algebraic groups is a variety not a scheme. I was getting confused because with schemes you look at spec not max spec and so for example the scheme of prime ideals of $ \mathbb{C}[x_1,\dots,x_n] $ has the zero ideal prime not maximal so non closed point so Zariski topology not $ T_1 $. – Ian Gershon Teixeira Jan 21 '22 at 15:04

1 Answers1

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Finite subgroups are always contained in the maximal compact subgroup so if $ G $ has a finite maximal closed subgroup then $ G $ must be compact. Also a maximal subgroup always includes the center so if $ G $ has a finite simple subgroup which maximal then $ G $ must have trivial center in addition to being compact. Thus the only groups which have finite maximal closed subgroups are adjoint groups like $ SO(2n+1), PSO(2n),PU(n) $

Note that a finite simple group $ \Gamma $ is a maximal closed subgroup of $ PU_n $ if and only if the central extension $ n.G $ is a unitary $ 2 $ design as a subgroup of $ SU_n $. See Claim 3 of https://math.stackexchange.com/a/4477296/758507

Some examples of finite simple groups appearing as subgroups of $ PU_n $ are given here

https://mathoverflow.net/questions/414265/alternating-subgroups-of-mathrmsu-n

here

https://mathoverflow.net/questions/414315/finite-simple-groups-and-mathrmsu-n

the references in

https://mathoverflow.net/questions/17072/the-finite-subgroups-of-sun

and here

https://mathoverflow.net/questions/344218/on-the-finite-simple-groups-with-an-irreducible-complex-representation-of-a-give

But when it comes to the maximality of such finite simple subgroups of $ PU_n $ the most useful reference is

https://arxiv.org/abs/1810.02507

which, read correctly, supplies a full classification of maximal closed subgroups of $ PU_d $ that happen to be finite and simple.

The classification consists of a few infinite families of examples of maximal closed subgroups of $ PU_d $ which are finite and simple

$ PU_d $, $ d=\frac{3^k -1}{2} $ and $ d=\frac{3^k +1}{2} $ both have a maximal $ PSp_{2k}(3) $ for $ k \geq 2 $.

$ PU_d $, $ d=\frac{2^k-(-1)^k}{3} $ has a maximal $ PSU_k(2) $ for $ k \geq 4 $

In addition to these, there are a few dimensions $ d $ for which $ PU_d $ has more maximal closed finite simple subgroups than we would expect. These exceptional case are:

$ PU_2 $: $ A_5 $

$ PU_3 $: $ A_6,GL_3(2) $

$ PU_4 $: $ A_7,PSU_4(2) $

$ PU_6 $: $ A_7,PSL_3(4), PSU_4(3) $

$ PU_8 $: $ PSL_3(4) $

$ PU_{10} $: $ M_{11}, M_{12} $

$ PU_{12} $: $ Suz $

$ PU_{14} $: $ ^2 B_2(8) $

$PU_{18} $: $ J_3 $

$PU_{26} $: $ ^3 F_4(2)' $

$PU_{28} $: $ Ru $

$PU_{45} $: $ M_{23},M_{24} $

$PU_{342}$: $ O'N $

$PU_{1333}$: $ J_4 $

All these finite simple maximal closed subgroups of $ PU_n $ lift to finite quasisimple maximal closed subgroups. Some of these quasi simple lifts have simple "section" so to speak and thus correspond to a finite simple maximal closed subgroups of $ SU_n $. Some examples are

$ SU_3 $: $ GL_3(2) $

$ SU_6 $: $ A_7 $

Ian Gershon Teixeira
  • 10,176