Writing a permutation group in 2 row notation

Question

I have a permutation group in $S_7$, namely: $$(12345)(137)(56)$$

How do I write this in two row notation? I am to write it as disjoint cycles and then as transpositions but I feel better working in two row notation.

Answer 1

Just compute where each element goes:

$1\mapsto3\mapsto4$

$2\mapsto3$

$3\mapsto7$

$4\mapsto5$

$5\mapsto6$

$6\mapsto5\mapsto1$

$7\mapsto1\mapsto2$.

So you have $$\begin{pmatrix} 1&2&3&4&5&6&7\\ 4&3&7&5&6&1&2 \end{pmatrix}$$

Answer 2

First, do the product as cycles and then write it as a matrix:

$$(12345)(137)(56)=(1456)(237)\longrightarrow\binom{1234567}{4375612}$$

Answer 3

It should be $$\binom{1234567}{4375612}$$ And you compose from right to left