Exercises

1. In $S_5$ perform the indicated operations; write each of the following in the 2-row form (2.1):

   1. $(1,2,3) (1,3) (1,4,5) (1,2)$,
   2. $(1,3,4)^{-1} (1,2) (1,5,3,2)$,
   3. $(1,3)^{-1} (1,2,4,5) (1,3)$.

2. Determine all the elements of $S_4$. Write each such permutation as a product of disjoint cycles.

3. Determine all subgroups of $S_3$.

4. Let the permutation $f$ be a cycle of length $k \geq 1$ (called a $k$-cycle). Show that $o(f) = k$.

5. Let $f = g_1g_2...g_r$ be the factorization of the permutation $f \in S_n$ into disjoint cycles $g_k$. If each $g_k$ is an $n_k$-cycle, $k = 1, 2, ..., r$, determine $o(f)$ and justify your answer.

   HINT: Use the result of problem 4.