Q66 / 170 · Manipal (MET) 2017 · Mathematics

If every element of a group $G$ is its own inverse, then $G$ is

  • A
    finite
  • B
    infinite
  • C
    not abelian
  • Correct answer
    D
    abelian

Solution

Let $a$ and $b$ be any element of group $G$. $\begin{aligned} & \therefore a=a^{-1}, b \geq b^{-1}, \forall a, b \in G \\ & \text { Now, }(a b)^{-1}=a b[\because a, b \in G \Rightarrow a b \in G] \\ & \Rightarrow b^{-1} a^{-1}=a b \Rightarrow b a=a b \forall a, b \in G \end{aligned}$ Hence, commutative law holds in group $G$. Therefore, $G$ is abelian group.

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

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