Algebraic Topology

1. add Prove that the groups $(2Z,+)$ and $(Z,+)$ are isomorphic.

   Define the function $\varphi :Z\to 2Z$ by $\varphi \left(n\right)=2n$ for all $n\in Z$. We will first show that $\varphi$ is a bijection.

   Assume that $\varphi \left(a\right)=\varphi \left(b\right).$ Then $2a=2b$ which implies $a=b.$ Hence, $\varphi$ is injective. Now, let $n\in 2Z$. Since $n$ is even, $n=2k$ for $k\in Z.$ Now, $\varphi \left(k\right)=2k=n$ and $\varphi$ is surjective. Since $\varphi$ is both injective and surjective, it is bijective.

   Lastly, we have $$\varphi (a+b)=2(a+b)=2a+2b=\varphi \left(a\right)+\varphi \left(b\right)$$

   Thus, $(2Z,+)$ and $(Z,+)$ are isomorphic.

   Q.E.D.