Isomorphism

An isomorphism in group theory is a bijective map (morphism) that preserves the group operation between two groups.
Bijective maps, or bijections, are functions that are both injective (they preserve distinctness: no two different input elements have the same output) and surjective (they are onto: every element of the output set is the output for some input).
Two groups that are isomorphic are essentially the “same” group in terms of their group structure, they just possibly use different sets or a different operation.

To be a morphism (which is needed for an isomorphism), a function must satisfy the condition f(xy) = f(x)f(y) for all x and y in the group. This means that mapping preserves the group operation.

Group isomorphisms preserve various types of group properties, including:
- Order: The isomorphism translates between the two groups in such a way that the number of elements (order) in each group is the same.
- Group Operation: The operation of the two groups connects through the isomorphism so that if a * b = c in the first group, then the isomorphism of a * the isomorphism of b equals the isomorphism of c in the second group.
- Identity Element: The isomorphism maps the identity element of the first group to the identity element of the second group.
- Inverses: The isomorphism maps inverses in the first group to inverses in the second group.

For example, consider the groups (Z, +), the integers under addition, and (2Z, +), the even integers under addition. The function f(x) = 2x is an isomorphism from Z to 2Z, since it preserves the group operation: f(m + n) = 2(m + n) = 2m + 2n = f(m) + f(n).
However, the function g(x) = x^2 from (R{0}, *), the non-zero real numbers under multiplication, to itself, is not an isomorphism as it does not preserve the group operation: g(xy) = (xy)^2 is not equal to g(x)g(y) = x^2 * y^2 in general.

Isomorphisms are used to show that two groups are functionally the same in terms of their group structure. This can simplify complex problems by relating them to simpler, well-studied groups.

Other forms of isomorphisms exist as well, depending on the conditions. Such as:
- Endomorphism: A morphism which maps a group to itself, it’s a special type of isomorphism.
- Automorphism: An isomorphism where the source group and destination group are the same.
- Homomorphism: A more general form of a morphism, where it is not required to be bijective, just to preserve the group operation.

Though enlightening, isomorphisms are not always easy to find, and proving that two groups are isomorphic can be challenging. Learning common patterns and methods can help in identifying potential isomorphisms.