Group Isomorphism

*** Let and be binary algebraic structures. A bijection is an isomorphism if

We say (i.e., the groups are isomorphic. * It is a bijective Group Homomorphism

A structural property is a property that is preserved under isomorphism.
(Fraleigh 8.15) Let and be groups and be a one-to-one function such that . Then the image is a Subgroup of and provides an isomorphism of with . In other words
An automorphism is an isomorphism of a group to itself (i.e., )
- The inner automorphism of by is defined as Performing it is called the conjugation of by
- A conjugate subgroup of is given by
As a corollary to (Fraleigh 13.18) a mapping can be shown to be an Isomorphism by showing the following
- is a homomorphism
- is onto.
(Fraleigh 34.5) Second Isomorphism Theorem Let and . Then

(see Normal Group)
- Idea: Let be the canonical homomorphism for such that We have the following commutative diagram:

Diagrammatic Proof Sketch of the Second Isomorphism Theorem. Original Image

(Fraleigh e34.9) Let with and . Then
(Fraleigh 34.7) Third Isomorphism Theorem. Let with . Then

(see Factor Group)
- Idea:

Diagrammatic Proof Sketch of the Third Isomorphism Theorem. Image taken from Fraleigh

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