Thread: Groups and Subgroups

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, +>.

Originally Posted by mous99

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, +>.

Verify (i) . (ii) If integers with , then with integers and .

Originally Posted by FernandoRevilla

Verify (i) . (ii) If integers with , then with integers and .

this is not enough. The last condition that remains is that every element in the subset has to have an additive inverse.

Originally Posted by rayman

this is not enough. The last condition that remains is that every element in the subset has to have an additive inverse.

Those are necessary and sufficient conditions, according to a well known therorem (characterization of subroups).

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.

and that completes the subgroup criterion

Originally Posted by rayman

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

For if you are interested, here is proof that those three conditions are equivalent to only two: PlanetMath

Originally Posted by FernandoRevilla

Verify (i) . (ii) If integers with , then with integers and .

How to verify (i)?

Originally Posted by mous99

How to verify (i)?

Choose for example, and . Then, .