Question

2.9. Let a and b be integers.

(a) Prove that the subset aZ + bZ = {ak + bl | l, k ∈ Z } is a subgroup of Z.

(b) Prove that a and b + 7a generate the subgroup aZ + bZ.

Answer 1

a) Clearly $a\mathbb{Z}+b\mathbb{Z}\ne\phi$. Let $ak_1+bl_1$, $ak_2+bl_2$ be two elements in $a\mathbb{Z}+b\mathbb{Z},$ where $k_i,l_i\in\mathbb{Z}$. We have $(ak_1+bl_1)-(ak_2+bl_2)=a(k_1-k_2)+b(l_1-l_2)\in a\mathbb{Z}+b\mathbb{Z}$ as $k_1-k_2,l_1-l_2\in\mathbb{Z}$. This implies $a\mathbb{Z}+b\mathbb{Z}$ is a subgroup of $\mathbb{Z}$.

b) Firstly $a,b+7a\in a\mathbb{Z}+b\mathbb{Z}$. Secondly, given any $ak+bl\in a\mathbb{Z}+b\mathbb{Z}$ we can write $ak+bl=a(k-7l)+(b+7a)l$ , this implies $a$ and $b+7a$ generate $a\mathbb{Z}+b\mathbb{Z}$.