$\{G,*\}$ is a group with identity element $e$, and $\{G',\circ\}$ is a group with identity element $e'$. Let $S=G\times G'$. Define the “product” of pairs of elements $(a,a'),(b,b')\in S$ by $(a,a')(b,b')=(a\circ b,a'*b')$
Prove that $S$ is a group under the “product” operation.
My first thoughts on the problem was that how can we prove that $a\circ b\in G$ and $a'*b'\in G'$ and thus that $S$ is closed under the "product" operation. The problem is, I can't seem to find a way to do so. The fact that $a*b\in G,\forall a,b\in G$ doesn't seem to give any information about whether $a\circ b\in G,\forall a,b\in G$. Any help with regards to closure would be helpful. Hopefully I can use that help to figure out associativity, identity and inverse for myself.