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Abstract Algebra (3th ed.)/Chapter1/Exercises 1.2

05.03.2020

1)

In general $|s| = 2$ by (2). And we can compute

  1. For $D_6$ we have the following elements $$ \{ 1,r,r^2 , s , sr , sr^2 \} $$ Now $|1| = 1, |r| = 3$. And we have that $$ (r^2)^2 = r^4 = r $$ So $|r^2| = 3$. $s(rs)r = ssr^{-1}r = 1$, so $|sr| = 2$. We have that $$ s (r^2 s) r^2 = ss r^{-2}r^2 = 1 $$ So $|sr^2| = 2$. In total
    1rr^{2}ssrsr^{2}
    133222
  2. For $D_8$ we have the following elements $$ \{ 1,r,r^2,r^3,s,sr,sr^2,sr^3 \} $$ here $|r| = 4$. We have that $(r^2)^2 = 1$. So $|r^2| = 2$. We have that $|r^3| = 4$. We have that every element on the form $sr^i$ has order $2$. That is $$ s(r^i s) r^i = ss r^{-i} r^{i} = 1 $$ So in total
    1rr^{2}r^{3}ssrsr^{2}sr^{3}
    14242222
  3. For $D_{10}$ we have the following elements $$ \{ 1,r,r^2,r^3,r^4,s,sr,sr^2,sr^3,sr^4 \} $$ We have $|r| = 5$. We have that $|r^2| = 5$. We have that $|r^3| = 5$. And $|r^4| = 5$. In total we get
    1rr^{2}r^{3}r^{4}ssrsr^{2}sr^{3}sr^{4}
    1555522222
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