Q134 / 200 · Manipal (MET) 2013 · Mathematics

If $(G, *)$ is a group such that $\left(a^* b\right)^2=(a * a)$ * $\left(b^* b\right)$ for all $a, b * G$, then $G$ is

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

Solution

$\left(a^* b\right)^2=\left(a^* a\right)^*\left(b^* b\right)$ for all $a, b \in G$ $\Rightarrow\left(a^* b\right)^*\left(a^* b\right)=\left(a^* a\right)^*\left(b^* b\right)$ for all $a, b \in G$ $a^*\left(b^* a\right)^{\star} b=a^*\left(\left(a^* b\right)^{\star} b\right.$ for all $a, b \in G$ $\Rightarrow b^* a=a^* b$ for all $a, b \in G$ \{by cancellation laws\} $\Rightarrow G$ is abelian.

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

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