Let $G$ be a finite group of order $n$. Let $M$ be a finitely generated abelian group generated by $m_1,m_2,・・・,m_k$.
Let $H^1(G,M)$ be a group cohomology.
Because $nH^1(G,M)=0$ from restriction and lcorestriction argument, and $H^1(G,M)$ is finitely generated, $H^1(G,M)$ turns out to be a finite group.
But what is the maximum of the order of $M$, denoted by $\#M$ ? How can we express $\#M$ in terms of $n$ and $k$ ?
P.S. My curiosity especially lies in the case $G=Gal(\Bbb{Q}(\sqrt{D})/\Bbb{Q}$) and $M=E(\Bbb{Q}(\sqrt{D}))$ where $E/\Bbb{Q}$ is an elliptic curve.$H^1(Gal(L/K),E(L))\cong H^1(Gal(\Bbb{Q}(\sqrt{D})/K),\Bbb{Z}^{rank(E/\Bbb{Q}(\sqrt{D}))}\times E(\Bbb{Q}(\sqrt{D})_{torsion}))$