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Dihedral Groups : Group Action

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Consider the "usual action " of Dihedral group of order 10
(D10) on the set {1,2,3,4,5}
and define D10 on the set of all 2-element subsets of {1,2,3,4,5}
by g*{i,j} ={g*i,g*j}
Find all the 2-element sets that are fixed by the element r i.e r*{i.j}={i,j}

I think there would be nothing, but i am sure .
So could someone give me full and cleat idea ?

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Solution Preview

The trick to unlocking your difficulty is to understand that
in a group action, individual elements can be swapped, but
the set as a whole can remain the same.

For example:-
5 2

Solution Summary

Group action is investigated. The solution is detailed and well presented.

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Burnside's Formula

There are 70 = C(8,4) ways to color the edges of an octagon, with four black and four white. The group D8 operates on this set of 70, and the orbits represent equivalent colorings. Count the number of equivalence classes.

D8 is the dihedral group associated with an octagon.

Please refer to the attachment for more details.

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