Question

If $\mathit{a}\mathbf{,}\mathit{b}\mathbf{\in }\mathit{G}$, prove that $\mathbf{\left|}\mathbf{a}\mathbf{b}\mathbf{\right|}\mathbf{=}\mathbf{\left|}\mathbf{b}\mathbf{a}\mathbf{\right|}$.

Short answer

If $a,b\in G$ then$\left|ab\right|=\left|ba\right|$ .

Step-by-step solution

Exercise 19

If $a,b\in G$then $\left|ba{b}^{-1}\right|=\left|a\right|$.

Proof of |ab|=|ba|

Let $a,b\in G$then,

$\begin{array}{c}ab=aba{a}^{-1}\\ =a\left(ba\right)a^{-1}\end{array}$

Find the order of ab as follows:

$\begin{array}{c}\left|ab\right|=\left|aba{a}^{-1}\right|\\ =\left|a\left(ba\right)a^{-1}\right|\\ =\left|a{a}^{-1}\right|\left|ba\right|\\ =\left|ba\right|\end{array}$

Therefore, if $a,b\in G$ then $\left|ab\right|=\left|ba\right|$.