Math 361, Spring 2022, Assignment 1

From cartan.math.umb.edu


Read:

  1. Section 14.

Carefully define the following terms, then give one example and one non-example of each:

  1. HG
  2. HG.
  3. Coset multiplication (when HG).
  4. Canonical projection (from G onto G/H).

Carefully state the following theorems (you do not need to prove them):

  1. Theorem characterizing when coset multiplication is well-defined.
  2. Theorem concerning the properties of coset multiplication ("When HG, coset multiplication turns G/H into a...").
  3. Theorem describing the kernel of the canonical projection.
  4. Fundamental Theorem of Homomorphisms.

Solve the following problems:

  1. Make an operation table for the quotient group Z12/4.
  2. 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 gG, one always has gG=e; now apply similar reasoning to the group G/H), and 31.
  3. 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.)
  4. Repeat the above exercise for the subgroup δ1 generated by δ1 alone.
  5. Let π:Z12Z12/4 denote the canonical projection. Write the table of values for π.
  6. 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.)
  7. 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.
  8. Recall that R denotes the set of non-zero real numbers, regarded as a group under ordinary multiplication, and let ϕ:SnR 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.
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Questions:

Solutions:

Complete Notes:

https://drive.google.com/file/d/1wCXbcBMd8br--G0xReIPzf8FN9sTtfHU/view?usp=sharing

Definitions:

  1. HG: H is a subgroup of G.
  2. HG: H is a normal subgroup of G. (When l,H and r,H are the same relation: if gG,hH,ghg1H
  3. Coset multiplication (when HG):(g1H)(g2H)=(g1g2)H.
  4. Canonical projection (from G onto G/H): Suppose HG. Define π:GG/H as follows: π(a)=aH. Then π is an epimorphism.

Theorems:

  1. Theorem characterizing when coset multiplication is well-defined: If H is a normal subgroup of G, Then coset multiplication is well-defined.
  2. Theorem concerning the properties of coset multiplication ("When HG, coset multiplication turns G/H into a..."): If HG, then G/H is a group under coset multiplication.
  3. Theorem describing the kernel of the canonical projection: Let π:GG/H be the canonical projection. Then ker(π)=H.
  4. Fundamental Theorem of Homomorphisms: Suppose ϕ:GH is a homomorphism. Let ϕ:GG/Ker(ϕ) be the canonical projection. Then there exists a unique monomorphism ˆϕ:G/Ker(ϕ)H such that ˆϕπ=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,amH.

Book Problem 31. kK, K is the intersection of all normal subgroups. gkg1 all normal subgroups, therefore, gkg1 intersection of all normal subgroups, gkg1K. 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. g1G,g2G,g1g2g11G because of closure. G/G:{G}