Paper 3, Section I, D

Let $g$ and $h$ be elements of a group $G$. What does it mean to say $g$ and $h$ are conjugate in $G$ ? Prove that if two elements in a group are conjugate then they have the same order.

Define the MÃ¶bius group $\mathcal{M}$. Prove that if $g, h \in \mathcal{M}$ are conjugate they have the same number of fixed points. Quoting clearly any results you use, show that any nontrivial element of $\mathcal{M}$ of finite order has precisely 2 fixed points.

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