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im stumped. i dont understand the 'does not contain a subgroup isomorphoc to Z_p X Z_p part.

ive tried using cauchy's theorem for abelian group: if G is a finite abelian group, and let p be a prime that divides the order of G. then g has an element of order p.

so, Z_p is a finite cyclic gruop, but Z_p X Z_p is not because p, p are not relatively prime.

what else.. :/

2 question: Let G be a group of odd order. prove that the mapping x --> x^2 from G to itself is one to one.

i know that the smallest group of odd order is {I, a, a^-1}. i dont understand the mapping, basically.