Proposition 1

— (i) Let $m \in M^G$ and $h \in H$. Then $hm \in M^{hGh^{-1}}$ and

(ii) Let F be a subgroup of G of finite index in G, and let $m \in M^F$, then

(iii) If $m \in M^H$, then $\mathrm{Tr}_{H/G} m = (H : G) \cdot m$.

(i) Let $h \in H$. For $h' \in H$ and $m \in M$ let us put $\varphi(h') = hh'h^{-1}$ and $\psi(m) = hm$. We have $\varphi(h') \psi(m) = \psi(h'm)$; by transfer of structure we deduce that if $m \in M^G$, then $hm \in M^{hGh^{-1}}$ and

Since $\mathrm{Tr}_{H/G}(m) \in M^H$, this proves (i).

(ii) Let $m \in M^F$ and let $(g_\alpha)_\alpha \in_A$ be a system of representatives of the left cosets of G mod F, and $(h_\beta)_\beta \in_B$ a system of representatives of the left cosets of H mod G. Then $(h_\beta g_\alpha)_{(\beta,\alpha) \in B \times A}$ is a system of representatives of the left cosets of H mod F. so

(iii) This assertion is evident.