An abelian group cannot be isomorphic to a non-abelian group due to the fundamental property that non-abelian groups exhibit non-commutative operations, while abelian groups are defined by commutative operations. The discussion references the Molecular Symmetry group, which is generally non-abelian, and contrasts it with point groups, which are typically abelian. This highlights a common misconception regarding the isomorphism between these types of groups. The character table of irreducible representations can be constructed from the multiplication table of the group, as discussed in group theory literature.
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DrDu said:In a non-abelian group, there are at least two elements A and B so that A*B=C but B*A=D with different D and C. this is not possible in an abelian group, so the two groups can also not be isomorphic.