Let where is a finite group.
If , what conditions we need in order to have ???
conjugacy class?? order??
Think about applying various post and pre multiplications of b inverse and c inverse. For example if we apply b inverse post multiplication on LHS we get a = cab^(-1) which is also equal to c^(-1)ab.
You can also get the identity in terms of a,b,c and b,c inverses.
This should give you a starting point to answering your question (you can use these identities to infer properties of a given that b = c).
So far, it appears that nobody has this right. While it is sufficient that G is abelian, or that a is in Z(G), it is not necessary. All that is required is that a and b commute, or (equivalently) a and c commute, nothing more.
To see a counter-example, let G = D4, with a = r3, b = c = r. None of these elements are in the center.