Answer to Question #290342 in Abstract Algebra for isara

Question

 Let A = {1,2,3,4,5,6} and let p1 = (3,6,2) and p2 = (5, 1, 4) be permutations of A.

(a) Compute p1 ◦ p2 and write the result as a product of cycles and as the product of transpositions.

(b) Compute p−1 ◦ p−1. 

Answer 1

$p_1=(362), p_2=(514)\\ p_1 \circ p_2 =(362) \circ (514)=\begin{pmatrix} 1 & 2&3&4&5&6 \\ 4 & 3 &6&5&1&2 \end{pmatrix}=(145)(236)$

b.)

$p_1^{-1}=(362)^{-1}=(263)\\ p_2^{-1}=(514)^{-1}=(154)\\ p_1^{-1} \circ p_2^{-1}=(263)\circ(154)=\begin{pmatrix} 1&2&3&4&5&6 \\ 5&6&2&1&4&3 \end{pmatrix}=(154)(263)$