Read: Continue with 4.5, and start 5.1.
Turn in: 4.116, 4.119 (only prove that \(A_n\) is a group), 4.120, 5.2, 5.3(a)
- For 4.116, make use of what you found in 4.97.
Remark about inverses in \(S_n\): it’s useful to know how to quickly compute the inverse of a permutation in cycle notation. First, just think about a single cycle like $\alpha = (1,4,5,7)$ and where $\alpha$ sends each number: \(1 \rightarrow 4 \rightarrow 5 \rightarrow 7 \rightarrow 1\). Now, where should $\alpha^{-1}$ send each number? Try to convince yourself that $\alpha^{-1}$ can be written by just reversing the cycle notation for $\alpha$ to get $\alpha^{-1}=(7,5,4,1)$. And if you want, you can rewrite $(7,5,4,1)$ as $(1,7,5,4)$.
Now, more generally, if $\beta$ is written as a product of cycles (disjoint or not), you can use Theorem 2.44 (“socks and shoes”) to see that $\beta^{-1}$ can be written by reversing the order of the cycles forming $\beta$ and then reversing the order of the numbers in each of the cycles (like we did with $\alpha$ above). For example, if $\beta = (1,4,5,7)(2,9)(3,6,8)$, then $\beta^{-1} = (3,6,8)^{-1}(2,9)^{-1}(1,4,5,7)^{-1} = (8,6,3)(9,2)(7,5,4,1)$.
Extra practice: 4.121