I've been set a maths problem, but I think that there might have been a mistake in it. It says, "Let G be a finite abelian group and let n:-|G|. Show that for some prime number q there is an element g of order q in G (Hint: Take any non-identity element h in g and show that some power of h has prime order).
My question is, how do you know that there is a non-identity element in the group? A set comprised of just the identity element along with some binary relation * will satisfy the group axioms, and the only element in this group has order 1. But then, as I have been taught that 1 is not prime, it appears that there is no element in this group that has prime order. (I can prove what is required assuming that there is some non-identity element in the group, as the question suggested, but surely it doesn't seem right to just make this assumption?)
2006-10-13
14:08:36
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3 answers
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friendly_220_284
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Mathematics