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.