Isomorphism of groups

Understanding Isomorphism of Groups

• An isomorphism of groups is a way of equating two groups so that their structure is preserved.
• Essentially, it involves a bijective function (that is, a function with an inverse) between two groups that respects the group operation.
• An isomorphism ensures that everything you can do in one group can be mirrored in the other. This concept is often used to study the underlying structure and behaviour of different groups.
• When two groups are isomorphic, we can safely say that they are effectively the ‘same’ group, only labelled differently.

Defining an Isomorphism

• A function f: G → H between two groups (G,) and (H,•) is an isomorphism if it is a bijective function and for all _a_ and _b_ in G, we have f(ab) = f(a) • f(b).
• The condition f(a*b) = f(a) • f(b) ensures that the group operation is preserved under the function.

Properties of Isomorphic Groups

• Two groups are considered isomorphic if their structure is identical. In essence, this means we can relabel one group to get the other.
• The order of the group is preserved under isomorphism. If group G has order n, then any group isomorphic to G also has order n.
• If one group is Abelian, then the group it is isomorphic to is also Abelian. This property is because the structure, including the commutative property, is preserved in an isomorphism.
• Importantly, isomorphic groups have the same group tables, up to renaming of elements.

Identifying Isomorphisms

• When given two groups, a good first step to identify an isomorphism is to look at the orders of the groups. If they are different, there cannot be an isomorphism.
• Another identifying feature is the order of elements. The isomorphism should preserve the order of elements within a group.
• General structure and operation rules in groups should also be maintained. For example, if one group has a certain identity element, the other group should have an equivalent.

Key Points to Remember about Isomorphism of Groups

• The concept of isomorphisms is central to many parts of mathematics as it allows the study of structures, rather than just individual objects.
• Isomorphisms help us classify groups. Two groups that seem different may turn out to be the ‘same’ from an algebraic point of view.
• One should always remember that being isomorphic is an equivalence relation: it is reflexive, symmetric, and transitive.

Applications of Isomorphisms of Groups

• In mathematics, isomorphisms are used not only in group theory but also in other areas like linear algebra, topology and graph theory.
• Outside mathematics, they have practical applications in physics, computer science, and various fields of engineering.

Reinforcing your understanding of the Isomorphism of Groups requires regular practice with different types of groups. Many examples and practice problems can help you familiarise yourself with these concepts.