So I was reflecting on the relationship between Gauss’s Lemma and Zolotarev’s Lemma in proofs of quadratic reciprocity:

GL: $ (a/p) = -1^n$ , where $ n$ is the number of least positive residues of $ ax$ for $ x$ in $ \{1, …, (p-1)/2 \}$ that are greater than $ p/2$

ZL: $ (a/p)$ = sign of the permutation in $ S_{p-1}$ induced by multiplying the numbers by $ a$

On the face of it, these feel like very different statements, so it’s almost surprising that both are true. One way to interpret this, I think, is to think of the partition of $ \{1, 2, …, p-1\}$ into two sets $ \{1, 2, …, (p-1)/2 \}$ and $ \{ (p+1)/2, …, p-1\}$ as an “indicator partition” for the sign of the permutation.

More formally, for a permutation $ \sigma$ in $ S_{p-1}$ , define an indicator partition to be a partition $ A \cup B = \{1, 2, …, p-1\}$ , $ A \cap B = \emptyset$ , if the parity of $ \sigma(A) \cap B = \epsilon(\sigma)$ .

For a set of permutations, $ X$ , in $ S_{p-1}$ , define an indicator partition to be a partition $ A \cup B = \{1, 2, …, p-1\}$ , $ A \cap B = \emptyset$ , if the parity of $ \sigma(A) \cap B = \epsilon(\sigma)$ for all $ \sigma$ in $ X$ .

Then the equivalence of the lemmas above is expressed by saying that $ \{1, 2, …, (p-1)/2 \}$ and $ \{(p+1)/2, …, p-1\}$ is an indicator partition for the set of $ p-1$ permutations defined by multiplication by $ a$ for $ a$ in $ \{1, 2, …, p-1\}$ .

Has this idea been explored? Are there interesting questions such as for any subgroup $ G$ of $ S_{p-1}$ , is there an indicator partition for $ G$ (or for which subgroups is there an indicator partition)?