Does G≅G/HG\cong G/H imply that HH is trivial?

Let G be any group such that

GG/H
where H is a normal subgroup of G.

If G is finite, then H is the trivial subgroup {e}. Does the result still hold when G is infinite ? In what kind of group could I search for a counterexample ?

Answer

Look at G=i=1 Z and the subgroup H=Z  i=2 0.

Attribution
Source : Link , Question Author : Klaus , Answer Author : Rasmus

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