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Math Help - Group with p order elements

  1. #1
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    Group with p order elements

    If G is an abelian group with order p^2, where p is prime.And for every elements in G have order p. Can I say that the group G is generated by 2 elements in G(every elements in G can be represented by the power of the product of a and b)??? I have tried the cases for p=2 but how about the others?? Could someone help me to prove it???
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  2. #2
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    Re: Group with p order elements

    If |G| = p2, and if every non-identity element has order p, then G cannot be cyclic, or else G would have an element of order p2.

    So pick any element a in G of order p, and let H = <a>. Then H is a non-trivial proper subgroup of G. since H is proper, there exists some b in G, but not in H. let K = <b>.

    Since G is abelian HK is a subgroup of G. Since p is prime, we have H∩K = {e}, thus |HK| = |H|*|K|/|H∩K| = p*p/1 = p2 so that HK = G, that is: G = <a,b>.
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