Permutation question concerning cycle shapes

  • #1
Confusedalways
4
0
Consider the subset of defined by



Show that for all and all , we have

I showed all the possible cycle shapes of h and am trying to show that must always have cycle shape , excluding the case of identity permutation.

Just don't know where to go from here
 
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  • #2
Hi, Confusedalways! Welcome. (Wave)

You're on the right track. Consider the following fact. If is a cycle in and , then . Take for instance . For all ,

so has cycle structure . The same argument applies to the others.
 

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