Paper 3, Section II, D

Groups | Part IA, 2010

Let GG be a group, XX a set on which GG acts transitively, BB the stabilizer of a point xXx \in X.

Show that if gGg \in G stabilizes the point yXy \in X, then there exists an hGh \in G with hgh1Bh g h^{-1} \in B.

Let G=SL2(C)G=S L_{2}(\mathbb{C}), acting on C{}\mathbb{C} \cup\{\infty\} by Möbius transformations. Compute B=GB=G_{\infty}, the stabilizer of \infty. Given

g=(abcd)Gg=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \in G

compute the set of fixed points {xC{}gx=x}.\{x \in \mathbb{C} \cup\{\infty\} \mid g x=x\} .

Show that every element of GG is conjugate to an element of BB.

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