Pure Mathematics E3: Homomorphisms

26 flashcards · Mathematics · by lakesare

1. [Isomorphisms] Visually imagine an automorphism.

   φ: (G, ◦) → (G, ◦) It is permissible for the domain group and the codomain group of an isomorphism to be the same group.Automorphism - isomorphism from G to G.

2. [Isomorphisms] What's the one-sentence meaning behind this isomorphism property?

   Renaming every el should result in a correct group table.

3. [Isomorphisms] What exactly does it mean when we say that φ preserves composites?

   These are the same: 1. x ◦ y 2. φ(x ◦ y) 1. compose x and y 2. take φ1. φ(x); φ(y) 2. φ(x) ∗ φ(y) 1. take φ 2. compose x and y

4. [Isomorphisms] If φ is an isomorphism from (G, ◦) to (H, ∗) - how could we find an isomorphism from (H, ∗) to (G, ◦)?

   φ–1

5. [Isomorphisms, homomorphisms] Isomorphism VS homomorphism?

   Isomorphism 1-1 and onto preserves composites homomorphism 1-1 and onto preserves composites A homomorphism that is both one-to-one and onto is an isomorphism.

6. [Isomorphisms] Visually imagine the "preserves composites" property.

   So φ(x ◦ y) goes first!

7. [Homomorphisms] What's a trivial homomorphism?

   A homomorphism that exists for any (G, ◦) → (H, ∗): x → eH. Given any two groups (G, ◦) and (H, ∗), there is always at least one homomorphism from (G, ◦) to (H, ∗), namely the mapping that maps every element of (G, ◦) to the identity element of (H, ∗).

8. [Isomorphisms, homomorphisms] Rate isomorhisms/homorphisms by how much structure they preserve.

   Isomorphism - preserves all structure. Homomorphism - preserves some structure. Trivial homomorphism - preserves very little structure.

9. [Homomorphisms] What exactly does it mean when we say that homomorphisms preserve conjugates.

   x, y are conjugate THEN φ(x), φ(y) are conjugate

10. [Homomorphisms] Why is linear transformation a homomorphism??

    Because of LT1! Remember that any vector space is, in particular, a group under vector addition. So a linear transformation maps from a group to a group. It is always a homomorphism between these groups, as shown below.

11. [Homomorphisms] What's Im φ?

    Image of φ. The image of a homomorphism

12. [Homomorphisms] In what case does a homomorphism preserves all structure?

    When it's one-to-one. Every one-to-one homomorphism φ preserves all of the structure of the domain group. This structure is preserved in Im φ, not necessarily in the whole codomain group. (It is preserved in the whole codomain group if φ is also onto, that is, if φ is an isomorphism.)

13. [Homomorphisms] How do you read Ker φ?

    Kernel of φ.

14. [Homomorphisms] Visually imagine Ker φ.

    The kernel of a homomorphism.The definition of the kernel of a homomorphism is essentially the same as the definition of the kernel of a linear transformation. You saw earlier that every linear transformation is also a homomorphism, so this is as you would expect.Note that the definition of a kernel applies only to homomorphisms, not to functions in general, because the codomain of a function need not contain an identity element. This contrasts with the definition of an image, which does apply…

15. [Homomorphisms] Visually imagine kernel of this φ.

    The identity element of the codomain group is 1. Hence Ker φ = { e, a, b, c }.

16. [Homomorphisms] Visually imagine these 3 theorems.

17. [Homomorphisms] What exactly justifies this claim?

    Given a group (G, ◦) and one of its normal subgroups K, there are many homomorphisms with domain group (G, ◦) and kernel K other than the one defined in the proof.

18. [Homomorphisms] What's the justification for this word: Kernormal.

    Ker φ ≡ normal subgroup!

19. [Homomorphisms] How do we immediately know this is a homomorphism?

    It's a trivial homomorphism!

20. [Kernel] What's similar between the behaviour of: the kernel kernel's cosets?

    the entire kernel maps to a single value the entire kernel's coset maps to a single value

21. [Kernel] Visually imagine what kernel cosets map to.

22. [First Isomorphism Theorem] What is the First Isomorphism Theorem generally about?

    It takes a homomorphism and creates an isomorphism out of that!

23. [First Isomorphism Theorem] Visually imagine the relationship between Ker and Im of a homomorphism. The First Isomorphism Theorem corollary.

24. [First Isomorphism Theorem vs Dimension Theorem] Relationship between Ker & Im of a homomorphism VS from abstract algebraDimension Theorem? from linear algebra

    First Isomorphism Theorem Dimension Theorem

25. [First Isomorphism Theorem] Visually imagine the First Isomorphism Theorem.

    Cosets of Ker(φ) ≅ Im(φ)

26. [First Isomorphism Theorem] How is this connected to the First Isomorphism Theorem?

    It tells us |G/Ker(φ)| = |Im(φ)|. Because the order of the isomorphic classes is the same.