Corollary 1

Let $(c_\sigma)_{\sigma \in \Gamma}$ be a family of non-zero elements of $N$. For $a \neq 0$ to exist in $N$ such that $c_\sigma = \sigma(a) \cdot a^{-1}$ for all $\sigma \in \Gamma$ it is necessary and sufficient that $c_{\sigma \tau} = c_\sigma \cdot \sigma(c_\tau)$ for $\sigma, \tau$ in $\Gamma$.