math.CO — Combinatorics

Distinct partial products in prime dihedral groups

Contributed by alexandre Thiery

Let $D_{2p}$ be the dihedral group of order $2p$, where $p$ is prime. For every fixed $0<\alpha<1$, we prove that every subset $A\subseteq D_{2p}\setminus\{e\}$ with $C_\alpha\le |A|\le p^\alpha$ admits an ordering whose nonempty partial products are pairwise distinct. Combined with the known small- and large-set theorems, this covers every subset size for all sufficiently large primes. Pairing reflections reduces the products to partial sums with fixed parity-labelled offsets. Two passes of block rotations remove distant collisions; bounded displacement allows the local condition to exclude the remaining collisions. The existence of suitable rotations follows from Pham--Sauermann's subset-sum anticoncentration theorem, an affine-rank estimate, and a count of clustered collision witnesses.