Exercises

1. Write the elements of both $S_3$ and $S_4$ as products of transpositions

   HINT: use the result of problem 2 from Section 3.1.

2. Prove that any element of $S_n$, $n > 1$, can be written as a product of transpositions of the form $(1,k)$ where $k = 2, ..., n$.

   HINT: First prove that for any transposition $(a,b)$, $(a,b) = (1,b) (1,a)(1,b)$.

3. Establish the two equations (3.5) and (3.6) used in the proof of Theorem 3.2.2.

4. Verify the case in the proof of Theorem 3.2.2 not done in the text. In particular, if $f \in S_n$ and $a,b$ occur in different cycles in the disjoint cycle representation of $f$, then use (3.6) to show $W((a,b)f) = W(f) + 1$.

5. Finally in the proof of Theorem 3.2.2, verify that

   (a) $m \equiv W(f)$ (mod $2$);

   (b) $W(f)$ is even if and only if $m$ is even.

6. Using the result of problem 1 above, determine both $A_3$ and $A_4$.

7. Prove that for $n > 2$, any element of $A_n$ can be written as a product of cycles of the form $(1,2,k)$, where $k = 3, 4, ... n$.

   HINT: Use the result of problem 2 and also establish that $(1,i,j) =(1,i,2)(1,2,i) (1,2,j)$ and $(1,j,2) = (1,2,j)^{-1}=(1,2,j)^2$.