Proving stuff is a Subgroup

1. Oct 12, 2008

SNOOTCHIEBOOCHEE

1. The problem statement, all variables and given/known data

Let a and b be integers

(a) Prove that aZ + bZ is a subgroup of Z+
(b) prove that a and b+7a generate aZ + bZ

2. Relevant equations

Z is the set of all integers

3. The attempt at a solution

(a)
In order for something to be a subgroup it must satisfy the following 3 properties:

(i)closure; that is that if aZ and bZ are in H (the subgroup of Z+) than aZ+bZ are in H.
(ii) identity: 0 is in H
(iii) inverses: if a(Z)+ b(Z) are in H, then so are -(a(Z) + b(Z)

i really dont know how to prove any of these are true for this particular subset.

I am also completley lost on part b.

Last edited: Oct 12, 2008
2. Oct 12, 2008

e(ho0n3

It's simpler than that: For H to be a subgroup of a group G, H must be nonempty and if a, b are elements of H, then ab-1 is an element of H.

Start by proving that aZ + bZ is not empty. Then pick two arbitrary elements x and y in aZ + bZ and show that x - y is in aZ + bZ.

Do you know what "a and b + 7a generate aZ + bZ" means?

3. Oct 12, 2008

SNOOTCHIEBOOCHEE

How do you show that aZ + bZ is non empty? it seems quite obvious to me but i dont know how to prove it.

Is it something like:

since a and b are integers, and when you mutiply integers with other integers, you get more integers.

I think that a and b+7a generate aZ+bZ means something like a cyclic subgroup generated by the elements a and b+7a is aZ + bZ

4. Oct 12, 2008

e(ho0n3

Give me a putative element of aZ + bZ and prove that it actually belongs to aZ + bZ.

And how does that relate to this problem.

Something like that. Can you provide more details.

5. Oct 12, 2008

SNOOTCHIEBOOCHEE

i know that the cyclic subgroup generated by a single element (lets say a) would be

(... a^-2 , a ^-1 , I , a, a^2, a^3, ...)

dunno how this works for 2 elements

6. Oct 12, 2008

SNOOTCHIEBOOCHEE

NM i figured this whole problem out

Thanks.

feel free to lock mods.