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If \((\mathrm{G}, *)\) is a group such that \((\mathrm{a} * \mathrm{~b})^2=(\mathrm{a} * \mathrm{a}) *\) \(\left(b^* b\right.\) ) for all a, b, \(\in \mathrm{G}\), then \(\mathrm{G}\) is
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abelian
\((\mathrm{a} * \mathrm{~b})^2=(\mathrm{a} * \mathrm{a}) *(\mathrm{~b} * \mathrm{~b})\) for all \(\mathrm{a}, \mathrm{b} \in \mathrm{G}\)
\(\Rightarrow(\mathrm{a} * \mathrm{~b}) *(\mathrm{a} * \mathrm{~b})=(\mathrm{a} * \mathrm{a}) *(\mathrm{~b} * \mathrm{~b})\) for all
\(\mathrm{a}, \mathrm{b} \in \mathrm{G}\)
\(\Rightarrow \mathrm{a} *(\mathrm{~b} * \mathrm{a}) * \mathrm{~b}=\mathrm{a} *(\mathrm{a} * \mathrm{~b}) * \mathrm{~b}\) for all \(\mathrm{a}, \mathrm{b} \in \mathrm{G}\)
\(\Rightarrow \mathrm{b} * \mathrm{a}=\mathrm{a} * \mathrm{~b}\) for all \(\mathrm{a}, \mathrm{b} \in \mathrm{G}\) (by cancellation laws)
\(\Rightarrow \mathrm{G}\) is abelian
\(\Rightarrow(\mathrm{a} * \mathrm{~b}) *(\mathrm{a} * \mathrm{~b})=(\mathrm{a} * \mathrm{a}) *(\mathrm{~b} * \mathrm{~b})\) for all
\(\mathrm{a}, \mathrm{b} \in \mathrm{G}\)
\(\Rightarrow \mathrm{a} *(\mathrm{~b} * \mathrm{a}) * \mathrm{~b}=\mathrm{a} *(\mathrm{a} * \mathrm{~b}) * \mathrm{~b}\) for all \(\mathrm{a}, \mathrm{b} \in \mathrm{G}\)
\(\Rightarrow \mathrm{b} * \mathrm{a}=\mathrm{a} * \mathrm{~b}\) for all \(\mathrm{a}, \mathrm{b} \in \mathrm{G}\) (by cancellation laws)
\(\Rightarrow \mathrm{G}\) is abelian
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