Let B := Im φ {\displaystyle {}B:=\operatorname {Im} \varphi } . We have e H = φ ( e G ) ∈ B {\displaystyle {}e_{H}=\varphi (e_{G})\in B} . Let h 1 , h 2 ∈ B {\displaystyle {}h_{1},h_{2}\in B} . Then there exist g 1 , g 2 ∈ G {\displaystyle {}g_{1},g_{2}\in G} such that φ ( g 1 ) = h 1 {\displaystyle {}\varphi (g_{1})=h_{1}} and φ ( g 2 ) = h 2 {\displaystyle {}\varphi (g_{2})=h_{2}} . Therefore, h 1 ⋅ h 2 = φ ( g 1 ) ⋅ φ ( g 2 ) = φ ( g 1 ⋅ g 2 ) ∈ B {\displaystyle {}h_{1}\cdot h_{2}=\varphi (g_{1})\cdot \varphi (g_{2})=\varphi (g_{1}\cdot g_{2})\in B} . Similarly, for h ∈ B {\displaystyle {}h\in B} there exists a g ∈ G {\displaystyle {}g\in G} fulfilling φ ( g ) = h {\displaystyle {}\varphi (g)=h} . Hence, h − 1 = ( φ ( g ) ) − 1 = φ ( g − 1 ) ∈ B {\displaystyle {}h^{-1}=(\varphi (g))^{-1}=\varphi (g^{-1})\in B} .
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