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\subsection*Exercise 7 State and prove the Orbit–Stabilizer Theorem.

Problems involving Group Actions and the Orbit-Stabilizer Theorem . dummit+and+foote+solutions+chapter+4+overleaf+full

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Let’s illustrate a complete answer as it would appear in your Overleaf document. dummit+and+foote+solutions+chapter+4+overleaf+full

"Show that $G$ acts on $X$ by [some rule]." dummit+and+foote+solutions+chapter+4+overleaf+full

\beginproof The group $G$ acts on itself by conjugation. The orbit of an element $x$ under this action is its conjugacy class, denoted $\mathcalO_x$ or $\textCl(x)$. The stabilizer of $x$ is the centralizer $C_G(x) = \g \in G \mid gxg^-1 = x\$.