The problem involves group theory, specifically examining a group G of order 12 and its properties related to normal subgroups and isomorphism with the alternating group A4. The original poster seeks to prove that a certain element b is in the center of G and to establish an isomorphism between G and A4 under specific conditions.
Participants have provided insights into the properties of normal subgroups and the structure of G, with some suggesting that the original poster clarify their earlier proofs. There is ongoing exploration of the implications of the center of G and the relationship between G and A4, with no explicit consensus reached yet.
The discussion highlights confusion regarding the order of elements and subgroups, particularly concerning the relationship between G and A4, as well as the implications of the center's order on the structure of G.