Q47 / 110 · VITEEE 2023 · Mathematics

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

  • Correct answer
    A
    abelian
  • B
    finite
  • C
    infinite
  • D
    None

Solution

\((\mathrm{a} * \mathrm{~b})^2=(\mathrm{a} * \mathrm{a}) *(\mathrm{~b} * \mathrm{~b})\) 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

Hence, the correct answer is option: 'A'.

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