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. The question was, Find a non-abelian group of order 21 (Hint: let a3=e and b7=e and find some i such that a-1ba=bi≠b which is consistent with the assumptions that a3=e and b7=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-1ba=bi≠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.