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thepreimageof1underisa\begin{array}{l}{\text { Let } G \text { and } H \text { be groups, and } \phi : G \rightarrow H \text { a group homomorphism.

 Let G and H be groups, and ϕ:GH a group homomorphism. If H1 is a  normal subgroup of H, prove that ϕ1(H1), the preimage of H1 under ϕ, is a  normal subgroup of G.

begin{array}{l}{text { Let } G text { and } H text { be groups, and } phi : G rightarrow H text { a group homomorphism. If } H_{1} text { is a }} \ {text { normal subgroup of } H, text { prove that } phi^{-1}left(H_{1}right), text { the preimage of } H_{1} text { under } phi, text { is a }} \ {text { normal subgroup of } G .}end{array}

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