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In Marder's Condensed matter physics, it uses matrix operations to explain how to justify two different lattice systems as listed in attachment.

However, I cannot understand why the two groups are equivalent if there exists a single matrix S satisfying S^{-1}RS^{-1}+S^{-1}a=R^{'}+a^{'}.

Can someone help me to understand it? Thank you.

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# Lattice systems and group symmetries

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