Isomorphisms
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Isomorphisms
The whole motivation for the study of abstract structures is in order to encapsulate what is common to several mathematical objects. Mathematicians study structure because the solution to a problem concerning one structure can be generalised to all other similar structures. Central to this idea of structure, then, is the concept of identity of structures. In the context of group theory we need to be able to demonstrate when two groups are essentially the same. In order to do this we will define the concept of an isomorphism. An isomorphism, [Equation goes here - download the original to see it.] is a structure preserving mapping (a mapping that maintains the structure) It maps the elements of one set G to elements of the other. In order to preserve structure the size of G and H must be identical. That is, an isomorphism is a one-one mapping (a bijection), of G to H. The binary operation O on G corresponds under the isomorphism to the binary operation on H. The isomorphism preserves the structure in G and shows that this structure is replicated in H. To do this we show that [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] This third property can be illustrated thus [Diagram goes here - download the original to see it.] There are two ways to navigate between the element [Equation goes here - download the original to see it.] and the group [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] The following diagram also illustrates this property [Diagram goes here - download the original to see it.] When two groups are isomorphic we use the symbol [Equation goes here - download the original to see it.] Summary An isomorphism [Equation goes here - download the original to see it.] Note, this second property is called the homomorphism property - it is the structure preserving mapping between the groups. We will learn subsequently that the structure can be preserved even if the mapping is not a bijection (one-one). When the structure is preserved but the mapping is not one-one, then the relationship established between the groups is a homomorphism.
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Contents of
Isomorphisms
1 Isomorphisms
2 When structures are not isomorphic
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