Subgroup of GL(2,$\mathbb C$)

Question

I need an example of a finite Group which is not isomorphic to a subgroup of GL(2,$\mathbb C$).

I know that every cyclic group is a subgroup but a concrete example of a finite group which is not a subgroup of GL(2,$\mathbb C$) is eluding my calculations. Please give one example if there's one. Thank you.

See here for the list of finite groups which arise as subgroups of $GL_2(\Bbb C)$. — Dietrich Burde, Aug 22 '20 at 16:56

score 3 · Accepted Answer · answered Aug 22 '20 at 17:14

3

$C_2\times C_2\times C_2$ is the smallest such group. To see this, note that abelian subgroups of $\mathrm{GL}_2(\mathbb{C})$ are diagonalizable. Thus if $G$ is an abelian subgroup of $\mathrm{GL}_n(\mathbb{C})$ then $G$ is generated by at most $n$ elements.

answered Aug 22 '20 at 17:14

David A. Craven

11,561

As one who does not know Representation theory of Groups, I would like to present the following points based on your answer. C_2\times C_2 belongs to GL(2,C) but not C_2\times\C_2\times\C_2. This is because there are 4 conjugacy classes of 2 by 2 invertible matrices available but 8 distinct conjugacy classes are not available. Please clarify if my understanding of your answer is correct. – Lawrence Mano Aug 22 '20 at 21:55
I'm afraid not. You only need 3 and 7 respectively, not 4 and 8, because the identity doesn't count. But anyway, not being conjugate inside the subgroup does not mean they are not conjugate in the group. The diagonal matrices with entries $(-1,1)$ and $(1,-1)$ are conjugate in the matrix group. They generate the group $C_2\times C_2$, and is up to conjugacy the unique subgroup $C_2\times C_2$ in $\mathrm{GL}_2(\mathbb{C})$. – David A. Craven Aug 22 '20 at 21:57
Thank you so much for clearing my doubts. I now have a good understanding of the problem. – Lawrence Mano Aug 22 '20 at 22:03

score 1 · Answer 2 · answered Aug 22 '20 at 16:11

1

Take a group with no nontrivial character of degree $\le2$, for instance $A_5$.

answered Aug 22 '20 at 16:11

Angina Seng

161,540

Thank you for the example. Can you please give some reference where are the proof of this could be found. – Lawrence Mano Aug 22 '20 at 16:14
Lots of textbooks do the $A_5$ character table as an example. – Angina Seng Aug 22 '20 at 16:15
1

As a point of interest, the group $A_5$ is a subgroup of $\mathrm{PGL}_2(\mathbb{C})$ though, just not $\mathrm{GL}_2(\mathbb{C})$. – David A. Craven Aug 22 '20 at 18:01

Subgroup of GL(2,$\mathbb C$)

2 Answers2