2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four

A ternary quasigroup (or 3-quasigroup) is a pair (N, q) where N is an n-set and q(x, y, z) is a ternary operation on N with unique solvability. A 3-quasigroup is called 2-idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x) = y. A conjugation of a 3-quasigroup, considered as an OA(3, 4, n), , is a permutation of the coordinate positions applied to the 4-tuples of . The subgroup of conjugations under which is invariant is called the conjugate invariant subgroup of . In this paper, we will complete the existence proof of the last undetermined infinite class of 2-idempotent 3-quasigroups of order n, n ≡ 1 (mod 4) and n > 9, with a conjugate invariant subgroup consisting of a single cycle of length four.