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Discussion on question by MathFail: For any finite group $G$, is it always true that $ b_0\cdot b_1 \cdot\cdot\cdot b_{n}=b_0?$

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Discussion on question by MathFail: For any finite group $G$, is it always true that $ b_0\cdot b_1 \cdot\cdot\cdot b_{n}=b_0?$