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$L=\{(ab)^n : n\text{ is a natural number apart from }6\}$, I want to show L is non-regular by finding an infinite set of L-distinguishable words. Could you help me?

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Well, you can't show that it is not regular, because it is regular.

Indeed, $L = (ab)^*\setminus \{(ab)^6\}$, and $(ab)^*$ is regular (concatenation + kleene star), and $\{(ab)^6\}$ is regular (because it is finite), and regular languages are closed under difference.

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