Two groups are isomorphic if there exists a bijective mapping (or function) between the elements of the two groups that preserves the group structure and operations.
To prove that two groups are isomorphic, you must show that there exists a bijective mapping between their elements that preserves the group structure and operations. This can be done by constructing an isomorphism, which is a function that satisfies these conditions, or by showing that all elements of one group can be uniquely mapped to elements of the other group.
Isomorphic groups have the same size or cardinality, the same group structure (e.g. commutativity, associativity, identity element), and the same group operations (e.g. multiplication, addition, exponentiation).
Yes, two groups can be isomorphic even if they have different names or notations. Isomorphism is a concept that focuses on the structure and operations of the groups, not their names or symbols.
Proving groups are isomorphic allows for the translation and transfer of knowledge between different groups. This can help in identifying patterns, solving problems, and making connections between seemingly unrelated groups. It also allows for the generalization of results from one group to another.