Hw7 Problem 2

### Problem 2

Write the following permutations as (a) a product of disjoint cycles; (b) a product of transpositions

(a) $(1235)(413)$

(b) $(13256)(23)(46512)$

(c) $(12)(13)(23)(142)$

Solution

(a) $\begin{pmatrix} 1&2&3&4&5&6\\ 5&3&4&2&1&6\end{pmatrix}$ Hence, the products are $(15)(234)$ and $(15)(24)(23)$.
(b) $\begin{pmatrix} 1&2&3&4&5&6\\ 2&4&5&1&3&6\end{pmatrix}$ Hence, the products are $(124)(35)$ and $(14)(12)(35)$.
(c) $\begin{pmatrix} 1&2&3&4\\ 4&3&1&2\end{pmatrix}$ Hence, the products are $(1423)$ and $(13)(12)(14)$.