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The question asks for me to write down the permutations on the set $\{1,2,3,4\}$ which are symmetries of the square with vertices as shown. Hence show that $D_4$ is a subgroup of $S_4$.

1 2

4 3

I have worked out the permutations, but confused on how to show that $D_4$ is a subgroup of $S_4$.

How do you define $D_4$ and $S_4$, and show that $D_4$ is a subgroup of $S_4$?

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The question is: How do you define D₄? –  k.stm Mar 13 '13 at 10:20
    
When $D_4$ is the set of "symmetries $\ldots\ $ as shown" then $D_4$ is a subgroup of $S_4$, whatever these symmetries are. –  Christian Blatter Mar 13 '13 at 14:39
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1 Answer

Hints:

i) The dihedral group $\,D_4\,$ contains either rotations (in integers multiples of $\,\pi/2\,$ , say around an imaginary pivot in the center of the square (= the intersection point of its diagonals), or reflections through one of the symmetry axis of the square (either a vertrical or a horizonal line through the square's middle or through one of the two diagonals).

ii) Example of rotation in an angle $\,\pi\,$ anti-clockwise: we get the following mappings of the vertices:

$$1\to 3\;,\;\;2\to 4\;,\;\;3\to 1\;,\;\;4\to 2$$

You can see that the above symmetry of the square is represented by the permutation $\,(13)(24)\,$ (written as product of disjoint cycles)

iii) Example of reflection, say through the diagonal $\,13\,$ in the square:

$$1\to 3\;,\;\;2\to 4\;,\;\;3\to 3\;,\;\;4\to 2$$

represented by the permutation $\,(24)\,$ ... etc.

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