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Set closed under multiplication and reflection mod p

Source: ICMC 2022 Round 1 P4

December 4, 2021
reflectionGroupsnumber theorymod pcollege contestsICMC

Problem Statement

Let pp be a prime number. Find all subsets S⊆Z/pZS\subseteq\mathbb Z/p\mathbb Z such that 1. if a,b∈Sa,b\in S, then ab∈Sab\in S, and 2. there exists an r∈Sr\in S such that for all a∈Sa\in S, we have r−a∈S∪{0}r-a\in S\cup\{0\}.
Proposed by Harun Khan
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