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Question Number 74840 by aliesam last updated on 01/Dec/19

Commented by kaivan.ahmadi last updated on 01/Dec/19

$$a^4=1\Rightarrow$$$$b^4=a^4=1\Rightarrow o\left(b\right)=4$$$${\left(a^2\right)}^2=1\Rightarrow o\left(a^2\right)=2$$$$o\left(a^3\right)=o(a.a^2)=\frac{o\left(a\right)o\left(a^2\right)}{(o\left(a\right),o\left(a^2\right))}=\frac{2\times 4}{2}=4$$$$o\left(ab\right)=\frac{o\left(a\right)o\left(b\right)}{(o\left(a\right),o\left(b\right))}=\frac{4\times 4}{4}=4$$$$o\left(a^{2}b\right)=\frac{2\times 4}{2}=4$$$$o\left(a^{3}b\right)=\frac{4\times 4}{4}=4$$

Commented by kaivan.ahmadi last updated on 01/Dec/19

$$a.a^3=1\Rightarrow a^{-1}=a^3,{\left(a^3\right)}^{-1}=a$$$${\left(a^2\right)}^2=1\Rightarrow {\left(a^2\right)}^{-1}=a^2$$$$\left(a^{2}b\right)b=a^2{b}^2=b^2{b}^2=b^4=1\Rightarrow {\left(a^{2}b\right)}^{-1}=b,b^{-1}=a^{2}b$$$$$$

Commented by kaivan.ahmadi last updated on 02/Dec/19

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