So I was working through some problems in Herstein's Algebra on my own time, and I came across something I wasn't so sure about.(adsbygoogle = window.adsbygoogle || []).push({});

The question was, Find a non-abelian group of order 21 (Hint: let a^{3}=e and b^{7}=e and find some i such that a^{-1}ba=b^{i}≠b which is consistent with the assumptions that a^{3}=e and b^{7}=e)

All the solutions say if we set i=2, then this generates a group of order 21. I was just wondering how exactly I could check this without using the semiproduct stuff, which I haven't yet learned. I understand that consistent means that it doesn't contradict the assumptions, but I'm exactly sure how to check that, and what constitutes a full checking that there is no contradiction. Taking a^{-1}ba=b^{i}≠b to all its powers and checking to see that they give no contradiction doesn't seem to be the right way to go about it.

Similarly, is there a quick way to say 3, 5 and 6 don't work (assuming I'm right that they don't)? I can check and find that they give a contradiction to the assumptions, but I don't know if this is the easiest way.

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# Non-Abelian Group of Order 21

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