Permutation Multiplication (easy)

Question

Given α◦β=(1532)(14)(35)

How do we get from the given to = ( 1 4 5 2 ) ( 3 ) = ( 1 4 5 2 ) = (4 1 3 5 2) ?

Thanks

Answer 1

It is easiest to write this out and compose from right to left as follows $$ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 5 & 1 & 2 & 4 & 3 \end{pmatrix}\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 3 & 1 & 5 \end{pmatrix}\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 2 & 5 & 4 & 3 \end{pmatrix} $$ $$ = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 1 & 3 & 5 & 2 \end{pmatrix} = (1452) $$ For instance, you check $1 \rightarrow 1 \rightarrow 4 \rightarrow 4$