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For my course we need to read a very old discrete maths book Discrete maths and its application (1995) and the explanation is just horrible.

Click here to see the picture

Here, point (4), it does not explain why (354)^3 = id, and why (354)^24 = id. I don't understand what id means, and what makes a permutation group id.

The permutations and groups chapter is no longer in the latest version (discrete mathematics and its applcation) in 2013. Do you know where I can find more information on this topic? I have been googling for a long time and I cannot find any resources on this topic.

Thanks.

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    id is the identity permutation. – 2017-01-29
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    @MarkBennet but wat makes a permutation identity permutation? :( Why is the power of permutation group (354) id? in the book it does not say anything at all. – 2017-01-29
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    Did you understand that $(354)^3$ means applying the permutation three times? – 2017-01-29
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    identity permutation is the permutation that doesn't move anything – 2017-01-29
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    1995=very old ? – 2017-01-29
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    The identity permutation sends each element of the underlying set to itself. Frankly, if you aren't comfortable with that, you really need to go back to the very beginning and make sure you understand everything before the example (and note it is giving examples not offering explanations, though it does expect that you will understand the examples being given). – 2017-01-29

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Any cycle of length $\ell$ has order $\ell$, i.e. if $\gamma= (a_1a_2\dotsm a_\ell)$, then $\gamma^\ell=\operatorname{id} $.

This results from the observation that, since the cycle notation denotes the mapping $$a_i\longmapsto a_{i+1\bmod\ell}\quad(i=1,2\dots,\ell),$$ then $\gamma^2$ is the mapping $a_i\longmapsto a_{i+2\bmod\ell}$. More generally, $\gamma^r$ is the mapping $$a_i\longmapsto a_{i+r\bmod\ell}\quad(i=1,2\dots,\ell).$$ Thus $\gamma(a_i)=a_i$ for all $i$ if and only if $i+r\equiv i\mod\ell$, i.e. if and only if $r\equiv 0\mod \ell$.