Z_N-Invariant Subgroups of Semi-Simple Lie Groups

We employ Mathematica to find $Z_N$-invariant subgroups of $E_8$ for application in M-theory. These $Z_N$-invariant subgroups are phenomenologically important and in some cases they resemble the gauge groups of our real world. We present a specific example of $Z_7$-invariant subgroups of $E_8$, which turn up in orbifold compactification of M-theory. However, the procedure can be applied for any $Z_N$ group that acts by shifts in the root lattice of semi-simple Lie groups.