The Preimage Of A Subgroup Under A Homomorphism Is A Subgroup

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Problem 100 (The Preimage Of A Subgroup Under A Homomorphism Is A Subgroup)

Suppose that $f:G\to H$ is a homomorphism. Let $B$ be a subgroup of $H$. Let $A$ be the preimage of $B$, namely $$A=f^{-1}(B)=\{g\in G\mid f(g)\in B\}.$$ Show that $A$ is a subgroup of $G$.

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