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Showing that a subgroup of Sym(4) is isomorphic to D_8

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Homework Statement


Let ##R## be the set of all polynomials with integer coefficients in the independent variables ##x_1, x_2, x_3, x_4##.
##S_4## acts on ##R## by the group action ##\sigma \cdot p(x_1,x_2,x_3,x_4) = p(x_{\sigma(1)},x_{\sigma(2)},x_{\sigma(3)},x_{\sigma(4)})##. Exhibit all permutations in ##S_4## that stabilize the element ##x_1x_2 + x_3x_4## and prove that they form a subgroup isomorphic to the dihedral group of order 8.

Homework Equations




The Attempt at a Solution


So, by listing the elements of ##S_4## and seeing how they act on ##x_1x_2+x_3x_4##, I found that the stabilizer of ##x_1x_2+x_3x_4## is ##\{1,(12),(34),(12)(34),(13)(24),(14)(23),(1324),(1423) \}##.

My problem is showing that this is isomorphic to the dihedral group of order 8. I think the correspondence is clear if we label the vertices of a square and consider the permutations as symmetries of the square, but I feel that exhibiting this 1-1 correspondence in that much detail is too computationally intensive and that there should be an easier way.
 

Answers and Replies

  • #2
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So, by listing the elements of ##S_4## and seeing how they act on ##x_1x_2+x_3x_4##, I found that the stabilizer of ##x_1x_2+x_3x_4## is ##\{1,(12),(34),(12)(34),(13)(24),(14)(23),(1324),(1423) \}##.

My problem is showing that this is isomorphic to the dihedral group of order 8. I think the correspondence is clear if we label the vertices of a square and consider the permutations as symmetries of the square, but I feel that exhibiting this 1-1 correspondence in that much detail is too computationally intensive and that there should be an easier way.
What is ##D_8\,##? And no, I'm not asking about the index, nor do I want to know that it is the dihedral group of order eight. Without this information, I'd say pin a screenshot of this Wiki page and you're done: https://de.wikipedia.org/wiki/Diedergruppe#Permutations-Darstellung
Just observe, that you have an odd numbering of the square: ##(12)## and ##(34)## are the diagonals and not neighboring points.
 
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