I have a small question about simple groups:
When we're decomposing a group $G$ we take a non-trivial normal subgroup of maximum order $G_1$ so that $\frac{G}{G_1}$ is simple and so on.
But why is $\frac{G}{G_1}$ simple?
I thought it had to do with the fact that $G_1$ is of maximum order, so there is no 'bigger' normal subgroup in $G$?