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)?
