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). 1. Le

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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).
1. Let eG and eH be the identities of G
and H, respectively. Show that φ(eG ) =
eH.
2. Let φ(G) = {φ(g)|g ∈ G}. Assume that φ is one-to-one. Show
that φ is an isomorphism if and only if φ(G) = H.
3. Let K be the set {g ∈ G|φ(g) = eH}. 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 = {eG }.