An isomorphism in abstract algebra is a bijective homomorphism between two algebraic structures. This means that the mapping preserves both the structure and the operations of the two structures being compared.
An automorphism is a special type of isomorphism where the two algebraic structures being compared are actually the same structure. In other words, an automorphism is an isomorphism between a structure and itself.
Isomorphism is important in abstract algebra because it allows us to compare and study different algebraic structures with similar properties and operations. It also helps us to understand the underlying structure of these structures and identify patterns and relationships between them.
To prove that two algebraic structures are isomorphic, you must show that there exists a bijective homomorphism between them. This can be done by demonstrating that the mapping preserves the structure and operations of both structures.
Yes, it is possible for two non-isomorphic structures to have the same cardinality. This means that they have the same number of elements, but the way these elements are combined and operated on may be different, making them non-isomorphic.