Math 361, Spring 2022, Assignment 1
From cartan.math.umb.edu
Read:
- Section 14.
Carefully define the following terms, then give one example and one non-example of each:
- H≤G
- H⊴G.
- Coset multiplication (when H⊴G).
- Canonical projection (from G onto G/H).
Carefully state the following theorems (you do not need to prove them):
- Theorem characterizing when coset multiplication is well-defined.
- Theorem concerning the properties of coset multiplication ("When H⊴G, coset multiplication turns G/H into a...").
- Theorem describing the kernel of the canonical projection.
- Fundamental Theorem of Homomorphisms.
Solve the following problems:
- Make an operation table for the quotient group Z12/⟨4⟩.
- Section 14, problems 1, 9, 24, 30 (hint: in December we used Lagrange's Theorem to prove that for any finite group G and any g∈G, one always has g∣G∣=e; now apply similar reasoning to the group G/H), and 31.
- Consider the group D4 of symmetries of a square; for purposes of this problem we will use the notation introduced on page 80 of the text. Let H denote the subgroup ⟨δ1,δ2⟩ generated by reflections in the diagonals. Determine whether H is a normal subgroup of D4. If it is a normal subgroup, then write the operation table for the quotient group D4/H. (Warning: normality can be checked by brute force, but this is very tedious and there is a shorter way. In the next problem you will actually need the brute-force check but it will be much shorter.)
- Repeat the above exercise for the subgroup ⟨δ1⟩ generated by δ1 alone.
- Let π:Z12→Z12/⟨4⟩ denote the canonical projection. Write the table of values for π.
- Let G be any group. Prove that the trivial subgroup {e} is normal in G. Then prove that the quotient group G/{e} is isomorphic to G itself. (Hint: you need to show that the canonical projection is injective in this case.)
- Let G be any group. Prove that the improper subgroup G is normal in G. Then write the operation table for the quotient group G/G.
- Recall that R∗ denotes the set of non-zero real numbers, regarded as a group under ordinary multiplication, and let ϕ:Sn→R∗ denote the sign homomorphism that takes even permutations to 1 and odd permutations to −1. Compute ker(ϕ), write an operation table for the quotient group Sn/ker(ϕ), and give a table of values for the momomorphism ˆϕ:Sn/ker(ϕ)→R∗ whose existence is asserted by the Fundamental Theorem.
Questions:
Solutions:
Complete Notes:
https://drive.google.com/file/d/1wCXbcBMd8br--G0xReIPzf8FN9sTtfHU/view?usp=sharing
Definitions:
- H≤G: H is a subgroup of G.
- H⊴G: H is a normal subgroup of G. (When ∼l,H and ∼r,H are the same relation: if ∀g∈G,∀h∈H,ghg−1∈H
- Coset multiplication (when H⊴G):(g1H)(g2H)=(g1g2)H.
- Canonical projection (from G onto G/H): Suppose H⊴G. Define π:G→G/H as follows: π(a)=aH. Then π is an epimorphism.
Theorems:
- Theorem characterizing when coset multiplication is well-defined: If H is a normal subgroup of G, Then coset multiplication is well-defined.
- Theorem concerning the properties of coset multiplication ("When H⊴G, coset multiplication turns G/H into a..."): If H⊴G, then G/H is a group under coset multiplication.
- Theorem describing the kernel of the canonical projection: Let π:G→G/H be the canonical projection. Then ker(π)=H.
- Fundamental Theorem of Homomorphisms: Suppose ϕ:G→H is a homomorphism. Let ϕ:G→G/Ker(ϕ) be the canonical projection. Then there exists a unique monomorphism ˆϕ:G/Ker(ϕ)→H such that ˆϕ∘π=ϕ
Problems:
1. 0+⟨4⟩ = {0,4,8};1+⟨4⟩ = {1,5,9}⋯. operation table: A table of Z4.
Book Problem 1. Z6/⟨3⟩ ⟨3⟩={0,3}. Order =62=3
Book Problem 9. 4
Book Problem 24. 2, Z2
Book Problem 30. |G/H|=m,(aH)m=eH=H,(aH)m=(am)H,am∈H.
Book Problem 31. k∈K, K is the intersection of all normal subgroups. gkg−1∈ all normal subgroups, therefore, gkg−1∈ intersection of all normal subgroups, gkg−1∈K. K is a normal subgroup.
3. Yes, 4
4. yes, 2
5. π(0)=0+⟨4⟩,π(1)=1+⟨4⟩,π(2)=2+⟨4⟩,π(3)=3+⟨4⟩
6. aE=a,bE=b,aE=bE,a=b,a==b
7. ∀g1∈G,g2∈G,g1g2g−11∈G because of closure. G/G:{G}