moving $a^{n}$ past b it makes sense for me for this to become $a^{n/3}b $ rather than $a^{3n}b$ from the given relation. Am I missing something obvious here ?
2026-02-22 22:38:13.1771799893
How does the element $ ba^{n} $ become $a^{3n}b $ from the relation $ ab=ba^{3}$ of the group $ D_{4}$?
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$$bab=a^3\implies (bab)^n=ba^nb=a^{3n}\implies ba^n=a^{3n}b$$
Remember all the time that $\;b^{-1}=b\;$ ...