Question on groups-how do you know that there is a non identity element in the group?

Question

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?)

Answer 1

The statement should say: "for every q, prime, a divisor of n, there is an element g of order q." Since no prime divides 1, there are no elements of prime order.

Answer 2

since the numbers are not nuymbers but letters-you have and(n) (g) (q) so you need to add 1 to hte mix which is the element of the prime number. But if you set the them up in the matter according to the rules, it show bring out the andswer of g-1+q.1=g-q=1.

Answer 3

for once i can honestly say i couldn't tell you if the above was true speech or gobbldygook. I truly have non idea WTF ur talking about 7 this must .: put a rather disconcerting handicap on our disenspondulating

i wonder how much study it took for you guys to work to understand maths. i gave up retaining maths, even times tables years ago!