Question

Let G be the group of integers under the operation of addition,

and let H = {3k | k ∈ Z}. Is H a subgroup of G

Answer 1

Since $G$ is a group and $H$ is a subset of $G$ (all elements of $H$ are integers), it is enough to check that for any two elements $a,b\in H$ we have: $a+b\in H$. It is the subgroup criterion. Suppose that $a,b\in H$. It means that they can be presented as: $a=3k,b=3z,$ $k,z\in{\mathbb{Z}}$. Then, $a+b=3k+3z=3(k+z)$. As we can see from the form, $a+b\in H.$ Thus, $H$ is a subgroup of $G.$