On existence of AIP neofields of odd order (Q908966)

A neofield is a structure \((N,+,\cdot)\) with two binary operations such that \((N,+)\) is a loop (with 0 as neutral element), \((N^*,\cdot)\) with \(N^*:=N\setminus \{0\}\) is a group, and both distributive laws are valid. The author considers finite AIP-neofields (i.e., neofields in which \((N^*,\cdot)\) is commutative and the loop \((N,+)\) has the inverse property) and proves the following result: Let G be an even order abelian group such that every odd prime which divides its order is congruent to 1 modulo 6. Then there exists an AIP- neofield \((N,+,\cdot)\) with \(G\simeq (N^*,\cdot)\) iff a) the 2-Sylow subgroup of G is cyclic, and b) \(| G| \neq 14\) (mod 24).