Is SL$(n,\mathbb{F})$ simple group and non abelian tensor square of GL$(n,\mathbb{F})$.

Question

I know very well that SL$(n,\mathbb{F})$ is a perfect group.

Can we say that it is simple and how can we find the non abelian square of GL$(n,\mathbb{R})$?

I have searched for these on google but did not get any affirmative result. Please help me.

Answer 1

No in general $SL(n,\mathbb{F})$ is not simple, suppose $\mathbb{F}=\mathbb{R}$ then it has non trivial center (https://en.wikipedia.org/wiki/SL2(R)). I think directly we can not say something about exterior square of $GL(n,\mathbb{R})$, but it will be an infinite group that contains $SL(n,\mathbb{R})$.