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Problem (Isomorphisms Yield An Equivalence Relation On The Set Of All Groups)

We say that $A$ is isomorphic to $B$ and write $A\approx B$ if and only if there exists a bijection from $A$ to $B$. In this problem, you'll prove that $\approx$ is an equivalence relation on the set of all groups.

  1. Let $A$ be a group. Show that $A$ is isomorphic to $A$ by building an isomorphism from $A$ to $A$.
  2. Suppose that $A$ is isomorphic to $B$. Prove that $B$ is isomorphic to $A$.
  3. Suppose that $A$ is isomorphic to $B$ and suppose that $B$ is isomorphic to $C$. Use this to produce an isomorphism from $A$ to $C$.

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