Permutations in rotations and reflections

[cat.inthe.hat]

Hi all, I've been having difficulty with the following question.

Let P be a regular pentagon. Let R be the rotation of P by 72degrees anticlockwise and let F be the reflection of P in the vertical line of symmetry. Represent R and F by permutations and hence calculate: F R^2 F R F^3 R^3 F, expressing this first as a permutation and then as a symmetry of P.I think I've correctly worked out R as the cycle (15432) and F = (25)(34). I've written these as permutations however, I don't understand how to do the calculation asked for and what it means by 'expressing as a symmetry of P'.

Any ideas would be much appreciated. Thanks in advance!

 

[JSuarez]

The calculation is just the composition of the R and F permutations, in the specified order. In your case, you first rotate counterclockwise by 72 degrees (F), then reflect three times (here, you may use the fact that R²=I, where I is the identity), then rotate again by 3x72 degrees, etc.

The final expression should express a symmetry.

 

[cat.inthe.hat]

The calculation is just the composition of the R and F permutations, in the specified order. In your case, you first rotate counterclockwise by 72 degrees (F), then reflect three times (here, you may use the fact that R²=I, where I is the identity), then rotate again by 3x72 degrees, etc.

The final expression should express a symmetry.

So, I should work out the permutations for F, R^2, ... etc. and then multiply them all in the order stated. Is this what you're saying? (Sorry I didn't quite understand).

 

[JSuarez]

Yes, that's pretty much it.

 

[cat.inthe.hat]

Yes, that's pretty much it.

Ok, thank you. =)