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Quaternion group presentation

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Show that the quaternion group Q_2 = {+1, -1, +i, -i, +j, -j, +k, -k} has presentation <a,b|a^4 =1, a^2 = b^2, ab = ba^3>

I need a rigorous proof with explanations so that I can study and understand please. I have an exam on Thursday.

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Solution Summary

This solution is comprised of a detailed explanation to show that the quaternion group Q_2 = {+1, -1, +i, -i, +j, -j, +k, -k} has presentation <a,b|a^4 =1, a^2 = b^2, ab = ba^3>.

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Recall the following useful result:

Let X be a set and Y a set of reduced words on X and let G = <X | Y>.

If H is any group with H = <X>, and H satisfies all the relations (in Y), then ...

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