Number of Distinct Conjugate Subsets
Theorem
Let $G$ be a group.
Let $S$ be a subset of $G$.
Let $N_G \left({S}\right)$ be the normalizer of $S$ in $G$.
Let $\left[{G : N_G \left({S}\right)}\right]$ be the index of $N_G \left({S}\right)$ in $G$.
The number of distinct subsets of a $G$ which are conjugates of $S \subseteq G$ is $\left[{G : N_G \left({S}\right)}\right]$.
Proof
$S^a = S^b \iff S^{a b^{-1}} = S$ (reference to be determined).
That is, $S^a = S^b \iff a b^{-1} \in N_G \left({S}\right)$, which is equivalent to $a^{-1} \equiv b^{-1} \left({\bmod\, N_G \left({S}\right)}\right)$.
Thus we have a bijection between the class $\mathcal{C} \left({S}\right)$ of subsets of $G$ conjugate to $S$ and the left coset space $G / N_G \left({S}\right)$ given by $S^a \to a^{-1} N_G \left({S}\right)$.
Since $G / N_G \left({S}\right)$ has $\left[{G : N_G \left({S}\right)}\right]$ elements, the result follows.
Find the reference to $S^a \text{ equals } S^b \iff S^{a b^{-1} } \text{ equals } S$.
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Sources
- Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.9$: Exercise $5.16 \ \text{(ii)}$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 49$
