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ISI 2021 | PCB Mathematics | Question: 5

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Let $G$ be a group generated by $a$ and $b$ such that $\operatorname{ord}(a)=n, \operatorname{ord}(b)= 2$ and $a b=b a^{-1}$, where $n$ is a positive integer, $b \notin\langle a\rangle$ and ord $(x)$ denotes the order of the element $x$.

  1. Prove that for any positive integer $k, a b^{k}=b a^{-k}$.
  2. Let $H$ be a cyclic subgroup of $\langle a\rangle$. Show that $H$ is a normal subgroup of $G$.
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