Let G and H be groups, and suppose φ : G → H is operation-preserving (that is, φ(ab) = φ(a)φ(b) for all a, b ∈ G). Let K

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Let G and H be groups, and suppose φ : G → H is
operation-preserving (that is, φ(ab) = φ(a)φ(b) for all a, b ∈
G).
Let K be the set {g ∈ G|φ(g) = e_H}. In other words, K is the
set consisting of all elements of G mapped to the identity of H by
φ. Assume that φ is onto. Show that φ is an isomorphism if and only
if K = {e_G }.