Which is the correct converse for the pictured question (so you can see the exact wording)?
Converse One: Take the converse of the statement agb=cgd always implies that ab=cd. That is:
Suppose G is a group with the property that ab=cd always implies agb=cgd. Prove or disprove G is abelian.
This seems like a kind of silly statement, and I see no way to prove it. The only way to make ab=cd is if a=c and b=d or if a=b=c=d.
Converse Two: Take the converse of the question. That is:
Suppose that G is an abelian group. Prove that G has the property that agb=cgd always implies ab=cd.
Which is too easy to be part (b) of this question. If G is abelian and agb=cgd, then we have
Which means that G has that property.