$SL(6, \mathbb{C})$ is not solvable

Question

Prove that $SL(6, \mathbb{C})$ is not solvable

Generally I never proved that a group is not solvable...

I thought about showing it is isomorphic to other groups which are not solvable (such as $S_6$) but I couldn't find any way to do this...

Other way I thought is to prove that the elements of the composition series are not $\mathbb{Z}/p \mathbb{Z}$, but I'm not sure how to show this as well as...

Any ideas?

Answer 1

• A subgroup of a solvable group is itself solvable.
• You seem to know that $A_6$ is not solvable.
• The set $G$ of $6\times6$ matrices such that each row and column has five zeros and a single one is a subgroup of $GL_6$ (known as permutation matrices because when viewed as linear transformations of $\Bbb{C}^6$ they simply permute the coordinates). $G$ is actually isomorphic to $S_6$. Furthermore, the determinant of a permutation matrix is the sign of the underlying permutation ($+1$ for even permutation and $-1$ for odd).
• The intersection of $G$ and $SL_6$ is equal to $A_6$, so ____________

Answer 2

Compute its commutator group: it is itself. This is a perfect group.

Add: To prove the special linear group is perfect for $n>1$, it suffices you prove that the elementary matrices generate it and, second, that the elementary matrices are commutators. This requires some hypothesis either on the parameter $n$ or on the field, so pay attention to this.