So if G = Q8 = <a, b : a^4 = 1, b^2 = a^2, b^{-1}ab = a^{-1}>(adsbygoogle = window.adsbygoogle || []).push({});

I'm fine with the notion of the derived subgroup G' = <[g,h] : g, h in G>

(Where [g,h] = g^{-1}h^{-1}gh)

But I can't see why G' = {1, a^2}, I can only seem to get everything to be 1!? i.e. g = a, h = a^3 ===> a^{-1}a^{-3}aa^3 = 1 etc.

And given the conjugacy classes of Q8 are: {1}, {a^2}, {a, a^3}, {b, a^2b}, {ab, a^3b}

Is it a case of just checking an element g and an element h from each of the 5 conjugacy classes, not all 8 elements? i.e. if we check g = b, we don't need to check g = a^2b as well.

(Sorry if this should be in the homework area, but it's not a "homework" question, just a problem I've come across reading a textbook and is a more general derived subgroup problem too!)

Thanks in advance!!! :-)

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# Derived Subgroup (In particular Q8)

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