Originally Posted by

**hgd7833** If G is finite Abelian group and G (X)_Z Z/pZ={0} ( tensor of G with Z/pZ over Z )

for all primes p, then show that G = {0}. Does the result remain true if G is infinite ?

Since G is finite abelian then G= Z/n_1Z x Z/n_2Z x ... x Z/n_kZ

where n_k | n_k-1 | ... | n_2 | n_1

so {0}= G(X)Z/pZ = Z/dZ where d=gcd(n_k , p) for all primes

then d=1 and for each p there is one of n_i relatively prime to p

but can this imply G is zero ?

Also, I have no idea about the second assertion.