Proving Closure and Identity of aZ + bZ as a Subgroup of Z+

[SNOOTCHIEBOOCHEE]

Homework Statement

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

Homework Equations

Z is the set of all integers

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 don't know how to prove any of these are true for this particular subset.

I am also completley lost on part b.

 

[e(ho0n3]

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)

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.

i really don't know how to prove any of these are true for this particular subset.

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.

I am also completley lost on part b.

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

 

[SNOOTCHIEBOOCHEE]

How do you show that aZ + bZ is non empty? it seems quite obvious to me but i don't 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

 

[e(ho0n3]

How do you show that aZ + bZ is non empty? it seems quite obvious to me but i don't know how to prove it.

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

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

And how does that relate to this problem.

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

Something like that. Can you provide more details.

 

[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

 

[SNOOTCHIEBOOCHEE]

NM i figured this whole problem out

Thanks.

feel free to lock mods.