On Special Near Orthomorphisms

Let \((G, \cdot)\) be a group with identity element \(e\) and
with a unique element \(h\) of order \(2\). In connection with an
investigation into the admissibility of linear groups, one of the
present authors was recently asked if, for every cyclic group \(G\)
of even order greater than \(6\), there exists a bijection \(\gamma$
from \(G \setminus \{e, h\}\) to itself such that the mapping
\(\delta: g \to g \cdot \gamma(g)\) is again a bijection from
\(G \setminus \{e, h\}\) to itself. In the present paper, we
answer the above question in the affirmative and we prove the
more general result that every abelian group which has a cyclic
Sylow \(2\)-subgroup of order greater than \(6\) has such a partial
bijection.