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Let \(\displaystyle{\left(G,\cdot\right)}\) be a group and \(\displaystyle{f:G\longrightarrow G}\)

be a homomorphism which is onto \(\displaystyle{G}\) .

If \(\displaystyle{H\leq G}\) such that \(\displaystyle{[G:H]=n\in\mathbb{N}}\), then prove that

\(\displaystyle{[G:f^{-1}(H)]=n}\) .

Let \(\displaystyle{\left(G,+\right)}\) be an abelian group.

Find the cardinality of \(\displaystyle{\rm{Hom}(\mathbb{Z},G)}\).

Thank you mr. Demetres.

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