Classification of Starters

A starter in an odd order abelian group \( G \) is a set of unordered pairs \( S = \{\{s_i, t_i\} : 1 \leq i \leq \frac{|G| – 1}{2}\} \), for which \( \{s_i\} \cup \{t_i\} = G \setminus \{0\} \) and \( \{\pm(s_i – t_i)\} = G \setminus \{0\} \). If \( s_i + t_i = s_j + t_j \) holds only for \( i = j \), then the starter is called a strong starter. Only cyclic groups are considered in this work, where starters and strong starters up to order \( 35 \) and \( 37 \), respectively, are classified using an exact cover algorithm. The results are validated by double counting.