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As the question says I have to find two subgroups of $GL_4(\mathbb C)$ which are finite and two which are infinite. For each of them I have to also give some properties(abelian, normal, order). This is my first algebra course so I do have a lot of problems grasping the concepts.

I thought about:

  1. Identity Matrix
  2. $GL_4(\mathbb C)$ itself

this are both the trivial subgroups where the first might be finite and the second infinite.

  1. Group of orthogonal matrices $4 \times 4$
  2. $SL(4,\mathbb C)$ all $4 \times4$ matrices wih determinant 1.

The problem is that i think now I have 3 infinite and 1 finite.

The second problem is regarding the properties that I have to write. Abelian: I think 1) is, 2) isn´t, I don´t know about 3) and 4). Order and if normal or not, I am completely clueless.

Thanks for any help

Derek Holt
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hallomate87
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    For more finite groups, you should think of other finite groups you know of (cyclic groups, dihedral groups, permutation groups), and how they would be embedded in $GL_1$ or $GL_2$ etc. If you can figure that out, there is an obvious way to put any subgroup of $GL_2$ into $GL_4$ (block-diagonally). – Joppy Mar 16 '21 at 10:30
  • This is better now as before. Did you follow the comment there? – Dietrich Burde Mar 16 '21 at 10:59
  • Tip: Try to think about simpler examples first. Can you find non trivial subgroups for $ GL_1(\mathbb{C})=\mathbb{C}^*=\mathbb{C}\setminus{0} $? What would be their analogoue in $GL_4(\mathbb{C})$ – Ben Mar 16 '21 at 11:06

1 Answers1

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Finite subgroups of $GL_4(\Bbb C)$ and more generally of $GL_n(K)$ have been studied a lot, see for example the following posts:

Finite-order elements of $\text{GL}_4(\mathbb{Q})$

Finite Subgroups of $GL(n,\mathbb{C})$

Consider for example subgroups for $n=2$ generated by $$ \begin{pmatrix}\cos 2\pi/m &-\sin2\pi/m \\ \sin2\pi/m & \cos 2\pi/m\end{pmatrix} $$ and embed into $GL_4(\Bbb C)$.

As for proper non-trivial infinite subgroups, we have $SL_n(\Bbb C)$, and also the infinite cyclic subgroup $C_{\infty}$, which is abelian. The group $SL_n(\Bbb C)$ is semisimple, as an algebraic group:

The Radical of $SL(n,k)$

It is a normal subgroup of $GL_n(\Bbb C)$, because it is the kernel of the determinant homomorphism $$ \det\colon GL_n(\Bbb C)\rightarrow \Bbb C^{\times}. $$

The orthogonal group is not normal in $GL_n$, see here:

Is $O(n)$ normal in $GL(n)$?

Dietrich Burde
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  • I thank you for your answer. The problem is that regarding the finite subgroups it is still incredibly abstract for me. I read the two other questions but most of the concepts that come are concepts which we still haven't touched. And also embeding things in GL_4 then is absolutely not easy. I don't know if there is any very simple example which i can grasp.For the infinite ones I am a lot more clearer now.Anyway thank you – hallomate87 Mar 16 '21 at 12:41
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    Embedding is really easy, if you consider block matrices in $GL_4$ with blocks of size two on the diagonal (and otherwise zero entries), right? What is "abstract" about powers of the sine and cosine matrix? It gives a nice subgroup. Do examples, i.e., write down matrices yourself. Do something yourself. – Dietrich Burde Mar 16 '21 at 12:43