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Romania NMO 2023 Grade 12 P1

Source: Romania National Olympiad 2023

April 14, 2023
abstract algebragroup theorynumber theory

Problem Statement

Let (G,⋅)(G, \cdot) a finite group with order n∈N∗,n \in \mathbb{N}^{*}, where n≥2.n \geq 2. We will say that group (G,⋅)(G, \cdot) is arrangeable if there is an ordering of its elements, such that
G={a1,a2,…,ak,…,an}={a1⋅a2,a2⋅a3,…,ak⋅ak+1,…,an⋅a1}. G = \{ a_1, a_2, \ldots, a_k, \ldots , a_n \} = \{ a_1 \cdot a_2, a_2 \cdot a_3, \ldots, a_k \cdot a_{k + 1}, \ldots , a_{n} \cdot a_1 \}.
a) Determine all positive integers nn for which the group (Zn,+)(Z_n, +) is arrangeable.
b) Give an example of a group of even order that is arrangeable.
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