Problem:Let be an isomorphism of groups. Show that is an isomorphism.

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I said:

If is an isomorphism, is a bijection between G and G' that maps every x in G to an x' in G'. Since is a bijection, this map can be traced back from the x' in G' back to the x in G (that being the inverse of x'), using the inverse of the isomorphism function, .

How wrong am I?