Let G be an abelian group, not necessarily primary. Define a subgroup S of G to be pure subgroup if, for all m in Z, S intersection mG = mS.
Prove that if G is a p-primary abelian group, then a subgroup S of G is pure if and only if S intersection p^nS for all n>= 0.
p is prime number.
I don't know how to test if the expression "S intersection p^nS for all n>=0" is true or false. That is not a boolean expression. That expression describes a collection of sets. It does not suggest that the sets need to exist. It does not suggest that the sets need to be equal to other sets. It is just a collection of sets. The intersection operator is not a comparison operator. Neither is the ^ operator. You are not comparing S intersection p^nS to anything. So, since that set is neither true nor false, the expression "If G is a p-primary abelian group, then a subgroup S of G is pure if and only if S intersection p^nS for all n>=0." is not a sentence (it can not be evaluated to true or false).
Edit: I can guess at what you were trying to say. If is a -primary abelian group, then a subgroup is pure if and only if for all .
If that is the case, then the argument seems pretty self-evident. The first direction of the implication is by definition. Next, to show the reverse implication, assume you have for all and show that for all . That seems like a straightforward argument to me. Where are you running into issues?