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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!

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    $\begingroup$ 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 $r^2s\notin\langle s\rangle$. $\endgroup$ – Hagen von Eitzen Feb 10 at 16:37
  • $\begingroup$ @HagenvonEitzen I apologize for the lack of clarity! I meant $D_{8}$ has the group of symmetries on a regular $8$-gon. $\endgroup$ – numericalorange Feb 10 at 17:11

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