Perhaps the most heavily utilized tool in Chapter 4 solutions is the Orbit-Stabilizer Theorem. It states that if a group acts on a set , then for any
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\section*Section 4.2: Orbits and Stabilizers dummit+and+foote+solutions+chapter+4+overleaf+full
\beginproof $G_a$ contains identity and is closed under multiplication and inverses. For the second part: \[ h \in G_g\cdot a \iff h\cdot(g\cdot a) = g\cdot a \iff (g^-1hg)\cdot a = a \iff g^-1hg \in G_a \iff h \in g G_a g^-1. \] \endproof Perhaps the most heavily utilized tool in Chapter