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Exhibit Dih(8) as a subgroup of Sym(4)


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.11

Write out the cycle decomposition of the eight permutations in $S_4$ corresponding to the elements of $D_8$ given by the action on the vertices of the square shown in the following figure.


Solution: The “flip” permutation, $s$, corresponds to the permutation $(2\ 4)$. The “rotation” corresponds to $(1\ 2\ 3\ 4)$. Now every other permutation is generated by these two, so we have the following.$$1 = 1 $$ $$r = (1\ 2\ 3\ 4)$$ $$r^2 = (1\ 3)(2\ 4)$$ $$r^3 = (1\ 4\ 3\ 2)$$ $$s = (2\ 4)$$ $$sr = (1\ 4)(2\ 3)$$ $$sr^2 = (1\ 3)$$ $$sr^3 = (1\ 2)(3\ 4).$$ Note that these results are consistent with the fact that every nonpower of $r$ in $D_{2n}$ has order 2, and the only elements of order 2 in $S_m$ are products of disjoint 2-cycles.

Linearity

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