Published online by Cambridge University Press: 01 August 2003
Let $G(o)$ and $G(*)$ be two groups of finite order $n$, and suppose that each of the sets $\{u\in G;\ uo v=u*v$ for all $v\in G\}$ and $\{v\in G;\ uo v=u*v$ for all $u\in G\}$ has $n/2$ elements. Then $G(*)$ can be obtained from $G(o)$ by one of the two general constructions that are discussed in the paper.