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Problem 51 (Inverses In Groups)

Suppose that $G$ is a group with $a,b\in G$.

  1. Prove that the inverse of $a^{-1}$ is $a$.
  2. Prove that the inverse of $ab$ is $b^{-1}a^{-1}$.
  3. If $a_1,a_2,a_3,\ldots, a_n\in G$, state the inverse of $a_1a_2a_3\cdots a_n$. Use induction to prove your claim.
If you see yourself repeating an induction proof similar to what we've been doing, then you're on the right track.

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