Half-automorphism group of Chein loops

A bijection f of a loop L is a half-automorphism if \(f(xy) \in \{f(x)f(y), f(y)f(x)\}\), for any \(x, y \in L\). A half-automorphism is nontrivial when it is neither an automorphism nor an anti-automorphism. A Chein loop \(L = G \cup Gu\) is a Moufang loop constructed from a group G and an element u of order 2 outside G. In this paper, the half-automorphism group of finite Chein loops is described.