Let Q be the set of rational numbers and H = { 5m/7^n | m, n are integers and n > o}.
Determine whether H is subgroups of <Q, +>.
Not according to my textbook. I am not sure what ''well known theorem'' you are referring to but according to subgroup criterion that is in my textbook (Durbin ''modern algebra'') in order to investigate whether a given subset is a subgroup we need to see it these criterions are satisfied
1) set should not be empty
2) it should be closed with respect to the operation
3) every element of the set should have an inverse with respect to the operation
Rayman, it's commonly known as the one-step subgroup test. You'll notice that in Fernando's second statement, it has a minus, and not a plus.
In any case, it's easy enough to see that the additive inverse will be from -m.
i.e. $\displaystyle \frac{5m}{7^n} + \frac{5(-m)}{7^n} = 0$
For if you are interested, here is proof that those three conditions are equivalent to only two: PlanetMath