Cross-posted from MSE


<https://math.stackexchange.com/questions/4272017/finite-maximal-closed-subgroups-of-lie-groups>


$\newcommand{\G}{\mathcal{G}} \newcommand{\K}{\mathcal{K}} \DeclareMathOperator\SU{SU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\SO{SO}$Let $\G$ be a Lie group.


I am interested in finite maximal closed subgroups of $ G $.


I'm guessing that $ \G $ has a finite maximal closed subgroup if and only if $ \G $ is simple and compact. Does anyone have other examples of finite maximal closed subgroups?



Finite maximal closed subgroups of Lie groups
$\newcommand{\G}{\mathcal{G}}
\newcommand{\K}{\mathcal{K}} $Question: When does $ \G $ admit a finite maximal closed subgroup?


Answer : Must be one of the following two cases


1. $ \G $ is compact and simple
2. $ \G $ is not compact in which case $ \G $ cannot be connected and moreover the component group $ \G/\G^\circ $ does not preserve any nontrivial proper closed subgroup (see [comment](https://mathoverflow.net/questions/405926/finite-maximal-closed-subgroups-of-lie-groups#comment1040850_405926) from YCor about $ C\_5 \ltimes \mathbb{R}^2 $).


From now on I will confine myself to the case that $ \G $ is connected.


In other words I will consider the statement "A connected Lie group $ \G $ has a finite maximal closed subgroup $ G $ if and only if $ \G $ is compact and simple."


The first implication is true.


**Claim 1:
If a connected Lie group $ \G $ has a finite maximal closed subgroup $ G $ then $ \G $ must be compact and simple.**


Proof:
Let $ \G $ be a connected Lie group and $ G $ a finite maximal closed subgroup. Since $ G $ is finite then $ G $ is a compact subgroup of $ \G $ so must be contained in a maximal compact subgroup, call it $ \K $. But $ G $ is a maximal closed subgroup thus we must have that $ \K=\G $ (note that $ \K $ cannot equal $ G $ since $ \K $ is connected (the maximal compact of a connected group is always connected)). So $ \G $ must be compact. If $ \G $ is not simple then there exists some morphism
$$
\pi: \G \to \G\_i
$$
with positive dimensional kernel (here $ \G\_i $ is basically one of the semisimple factors of $ \G $). Then
$$
\pi^{-1}(\pi(G))
$$
is a closed positive dimensional subgroup containing $ G $, contradicting the fact that $ G $ is a finite maximal closed subgroup. Thus if a connected Lie group $ \G $ has a finite maximal closed subgroup then we can conclude that $ \G $ is simple.


However the reverse implication does not hold: $ SU\_{15} $ is an example of a compact connected simple Lie group with no finite maximal closed subgroups.


To see why this is the case it is important to note that


**Claim 2: For a compact connected simple Lie group $ \G $, $ G $ is a finite maximal closed subgroup of $ \G $ if and only if $ G $ is Ad-irreducible and $ G $ is a maximal finite subgroup of $ \G $.**


this follows from Corollary 3.5 of [Sawicki and Karnas - Universality of single qudit gates](https://arxiv.org/abs/1609.05780).


Since a finite subgroup of $ SU\_n $ is Ad-irreducible if and only if it is a unitary 2-design we have


**Claim 3: $ G $ is a finite maximal closed subgroup of $ SU\_n $ if and only if $ G $ is a maximal unitary 2-group in $ SU\_n $.**


By inspecting Theorem 3 of [Bannai, Navarro, Rizo, and Pham Huu Tiep - Unitary $t$-groups](https://arxiv.org/abs/1810.02507) one immediately determines that $ SU\_{15} $ has no finite maximal closed subgroups.


Some of the main examples of finite maximal closed subgroups of $ SU\_n $ include the normalizer in $ SU\_{p^n} $ of an extra-special group $ p^{2n+1} $. Here $ p $ is an odd prime. There is also a similar construction $ p=2 $. These are known as (complex) Clifford groups. Then there are infinite families of examples relating to the Weil module for $ \operatorname{PSp}\_{2n}(3) $ and another family related to $ U\_n(2) $. Plus many exceptional cases.


A similar normalizer construction to the above gives finite maximal closed subgroups of all the $ \operatorname{SO}(2^n) $ as normalizers of an extra-special group $ 2^{2n+1} $. This is known as the real Clifford group. For details about real and complex Clifford groups see [Nebe, Rains, and Sloane - Self-Dual Codes and Invariant Theory](https://doi.org/10.1007/3-540-30731-1).


