Isomorphism

Defining Isomorphisms

• An isomorphism is a special type of map between groups that preserves their structure.
• It is a bijective function, meaning that it is both injective (no two elements map to the same element) and surjective (all elements in the image group can be mapped to).
• If an isomorphism exists between two groups, they are known as isomorphic groups. This denotes that the two groups have essentially the same structure.
• The existence of an isomorphism implies that the two groups behave in the same way under their respective group operations.

Properties of Isomorphisms

• Under an isomorphism, the identity element maps to the identity element, and the inverse of an element maps to the inverse of the image of that element.
• If two elements in the first group are combined to form a third, the images of those two elements will combine in the same way in the second group.
• These properties ensure that the group structures are preserved between two isomorphic groups.

Autobiography and Endomorphisms

• An endomorphism is a map from a group to itself. When such a map is bijective, it is referred to as an automorphism.
• Automorphisms are isomorphisms from a group to itself. These are the group’s symmetry transformations - changes that preserve the structure.

Examples of Isomorphisms

• The complex conjugation is an automorphism on the group of non-zero complex numbers under multiplication. The map sends each complex number to its complex conjugate.
• The set of integers under addition is isomorphic to the set of even integers under addition. The isomorphism here is the doubling map that sends each integer n to 2n.
• Between the groups of real numbers under addition and positive real numbers under multiplication, the map exp: x ↦ e^x is an isomorphism. The map ln: x ↦ ln(x) is its inverse, demonstrating the bijective property of isomorphisms.