The discussion revolves around proving the isomorphism between a group G of order 12 and the alternating group A4. The user has established some properties of G, including the existence of a normal subgroup of order 3 and a commutative element of order 2. They are struggling with two main parts: proving that the element of order 2 is in the center of G and demonstrating that G is isomorphic to A4, particularly using Cayley's theorem. The conversation highlights the importance of understanding the structure of G and A4, noting that A4 is the only subgroup of S4 with order 12, which is crucial for establishing the isomorphism. The discussion concludes with the need to prove that any subgroup of order 12 must be A4, reinforcing the connection between the properties of G and A4.