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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 ?

© BrainMass Inc. brainmass.com March 21, 2019, 1:28 pm ad1c9bdddf
https://brainmass.com/math/group-theory/dihedral-groups-group-action-97172

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:-
1
/
5 2
/
...

Solution Summary

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

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