Let $G,H$ be two groups , let $h(G,H)$ be the number of all group homomorphisms from $G$ to $H$ and $i(G,H)$ be the no. of all injective group homomorphisms from $G$ to $H$ . I know the relation $h(G,H)=\sum_{N \lhd G } i (G/N , H)$ . Now let $H,K$ be finite groups , if for any finite group $G$ , $h(G,K)=h(G,H)$ holds , then is it true that $i(G,H)=i(G,K)$ for any finite group $G$ ?
2026-03-30 23:15:31.1774912531
Let $H,K$ be finite groups , if for any finite group $G$ , $h(G,K)=h(G,H)$ holds , then is it true that $i(G,H)=i(G,K)$ for any finite group $G$ ?
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