Question 64
Let $(G, \cdot)$ be a finite group of order $n \in \mathbb{N}^*$, with $n \ge 2$. We shall call the group $(G, \cdot)$ *arrangeable* if there is an ordering of its elements, such that
$$
G = \{a_1, a_2, \dots, a_k, \dots, a_n\} = \{a_1 \cdot a_2, a_2 \cdot a_3, \dots, a_k \cdot a_{k+1}, \dots, a_n \cdot a_1\}.
$$
a) Determine all positive integers $n$ for which the group $(\mathbb{Z}_n, +)$ is arrangeable.
b) Give an example of an arrangeable group of even order.
$$
G = \{a_1, a_2, \dots, a_k, \dots, a_n\} = \{a_1 \cdot a_2, a_2 \cdot a_3, \dots, a_k \cdot a_{k+1}, \dots, a_n \cdot a_1\}.
$$
a) Determine all positive integers $n$ for which the group $(\mathbb{Z}_n, +)$ is arrangeable.
b) Give an example of an arrangeable group of even order.