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Therefore, $ab^-1 \in G_x$, and $G_x$ is a subgroup of $G$. \endproof

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\begintheorem[Orbit–Stabilizer] Let $G$ act on $A$ and $a\in A$. Then $|\mathcalO_a| = [G : G_a]$, where $\mathcalO_a = \g\cdot a \mid g\in G\$. \endtheorem Therefore, $ab^-1 \in G_x$, and $G_x$ is a subgroup of $G$

In exercises requiring you to find the number of elements with a certain property, your first instinct should always be to define an appropriate group action and apply this theorem. 2. The Class Equation If you share with third parties, their policies apply

When asked to find the size of a conjugacy class or the number of elements with a certain property, identify the group and the set . Use the identity:

: Educators often suggest using these guides to check work rather than as a primary learning source, as many exercises are designed to build intuition through struggle.