Paper 3, Section I, D

Groups | Part IA, 2021

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

Define the Möbius group M\mathcal{M}. Prove that if g,hMg, 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 M\mathcal{M} of finite order has precisely 2 fixed points.

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