# Showing two series are not the same.

I want to show that the following two composition series are not the same:

$$D_{8}\triangleright \left \langle s,r^{2} \right \rangle \triangleright \left \langle s \right \rangle \triangleright (1)$$

$$D_{8}\triangleright \left \langle s,r^{2} \right \rangle \triangleright \left \langle r^{2}s \right \rangle \triangleright (1)$$

I got these series from its lattice.

If I show this, then I can conclude they are distinct series for $$D_{8}$$. But how would I show this? I am not very familiar with dihedral groups!

• To begin with, there are competing notations for the dihedral group. Is your $D_8$ a group of $8$ elements or the symmetry group of a regular $8$-gon? But to show "not same", it should suffice that $\langle r^2s\rangle \ne \langle s\rangle$, e.g., by showing $r^2s\notin\langle s\rangle$. – Hagen von Eitzen Feb 10 at 16:37
• @HagenvonEitzen I apologize for the lack of clarity! I meant $D_{8}$ has the group of symmetries on a regular $8$-gon. – numericalorange Feb 10 at 17:11