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The question is to write the permutation $(123)(16543)$ as disjoint cycles.

First attempt (The wrong one):

$(16543)=\begin{pmatrix} 1 & 2 &3 & 4& 5&6\\ 6&2&1&3&4&5 \end{pmatrix}$

$(123)=\begin{pmatrix} 1 & 2 &3 & 4& 5&6\\ 2&3&1&4&5&6 \end{pmatrix}$

Now:

$(123)(16543)=\begin{pmatrix} 1 & 2 &3 & 4& 5&6\\ 2&3&1&4&5&6 \end{pmatrix}\begin{pmatrix} 1 & 2 &3 & 4& 5&6\\ 6&2&1&3&4&5 \end{pmatrix} = \begin{pmatrix} 1 & 2 &3 & 4& 5&6\\ 6&3&2&1&4&5 \end{pmatrix}=(1654)(23)$

Second attempt (the correct one):

$(123)(16543)$ becomes:

$1\to2\to2: 1\to2$
$2\to3\to1: 2\to1$

The first cycle becomes $(12)$

$3\to1\to6: 3\to6$
$6\to5$
$5\to4$
$4\to3$

The second cycle becomes $(3654)$

And the end result becomes: $(12)(3654)$

My question is: What is wrong with the first attempt?

Oualid
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  • By the order of permutation composition, the first one should be the correct one. – player3236 Jan 27 '21 at 13:31
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    Note that different authors use different conventions when it comes to permutations and the order in which they are applied. Some authors apply compositions of permutations from left to right. Other authors apply compositions from right to left. Nothing is "wrong" with either attempt per se beyond that you followed one convention in the first and the other convention in the second. You should confirm which order your teacher/class uses and/or make it clear which convention you are intending to follow. – JMoravitz Jan 27 '21 at 14:02
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    https://math.stackexchange.com/questions/527514/composition-of-permutations-left-to-right-or-right-to-left – JMoravitz Jan 27 '21 at 14:03
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    And, in case it wasn't obvious from this example, composition of permutations is not commutative in general. $\pi\circ\tau \neq \tau\circ \pi$, so order matters – JMoravitz Jan 27 '21 at 14:07

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