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Rewrite $(3412)(245) \in S_4$ as a product of distinct cycles.

I've only ever been given permutations as distinct cycles, transpositions or the matrix notation so I have no idea where to start.

Casteels
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Alianne Rose
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1 Answers1

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You can rewrite the cycles in the "matrix notation" as follows $\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2&3&4&1&5 \end{pmatrix}$ and $\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1&4&3&5&2 \end{pmatrix}$

Then you apply the first permutation to the second (assuming composition order) i.e, reorder the first row of the second matrix to be the second row of the first. $\begin{pmatrix} 2 & 3 & 4 & 1 & 5 \\ 4&3&5&1&2 \end{pmatrix}$

Giving the final permutation: $\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4&3&5&1&2 \end{pmatrix}$ Can you read off the cycles?

Ainsley Pullen
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