# Cosets II

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# Recall

Let G be an arbitrary group, and let H be a subgroup of G, H ≤ G. A left coset of H in G is a subset of the form aH = {ah | h ∈ H} for some a ∈ G. The collection of left cosets is denoted by G/H. Analogously, Ha = {ha | h ∈ H} is the right coset of H in G. The collection of right cosets is denoted by H\G.

Lemma. Let G be a group, let H be a subgroup of G, H ≤ G, and let g1, g2 ∈ G. The following statements are equivalent:

1. g1H = g2H
2. Hg1-1 = Hg2-1
3. g1H ⊆ g2H
4. g1 ∈ g2H
5. g1-1g2 ∈ H.

Proof: 1 ⇒ 2, let’s suppose g1H = g2H

∀x ∈ Hg1-1 ⇒ ∃h ∈ H, x = hg1-1 ⇒ x-1 =[The Socks and Shoes Principle] (g1-1)-1h-1 = g1h-1 ⇒ [x-1 = g1h-1, x-1 ∈ g1H = g2H] ∃h’ ∈ H: x-1 = g2h’ ⇒ x =[The Socks and Shoes Principle] h’-1g2-1 ∈ Hg2-1 ⇒ Hg1-1 ⊆ Hg2-1

Mutatis mutandis, the same reasoning applies to Hg2-1 ⊆ Hg1-1, and therefore Hg1-1 = Hg2-1

2 ⇒ 3) Let’s suppose Hg1-1 = Hg2-1. ∀x ∈ g1H, ∃h ∈ H, x = g1h ⇒ x-1 =[The Socks and Shoes Principle] h-1g1-1 ∈ Hg1-1 = Hg2-1 ⇒ ∃h’ ∈ H: x-1 = h’g2-1 ⇒ x =[The Socks and Shoes Principle] g2h’-1 ∈ g2H. Therefore, ∀x ∈ g1H, x ∈ g2H ⇒ g1H ⊆ g2H∎

3 ⇒ 4) Let’s suppose g1H ⊆ g2H ⇒ g1 = g1e ∈ g1H ⊆ g2H, and therefore g1 ∈ g2H∎

4 ⇒ 5) Let’s suppose g1 ∈ g2H ⇒ ∃h ∈ H: g1 = g2h ⇒ g1-1g2 = (g2h)-1g2 =[The Socks and Shoes Principle] (h-1g2-1)g2 =[Associativity] h-1(g2-1g2) = h-1 ∈ H ⇒ g1-1g2 ∈ H∎

5 ⇒ 1) Let’s suppose g1-1g2 ∈ H, ∀x ∈ g1H, ∃h, h’ ∈ H: x = g1h, g1-1g2 = h’ ⇒ h’-1 = (g1-1g2)-1 =[The Socks and Shoes Principle] g2-1g1 ⇒[h’-1 = g2-1g1 and multiplying both sides by g2, g2h’-1 = g2(g2-1g1)] g1 = g2h’-1 ⇒ x = g1h = g2h’-1h ∈ g2H ⇒ g1H ⊆ g2H

∀x ∈ g2H, ∃h, h’ ∈ H: x = g2h, g1-1g2 = h’ ⇒ g2 = g1h’ ⇒ x = g2h = g1h’h ∈ g1H ⇒ g2H ⊆ g1H ⇒[We have previously demonstrated that g1H ⊆ g2H] g1H = g2H∎

# Examples

• Let G = $GL(2, ℝ) = \bigl\{ {{[\bigl(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\bigr)]: a, b, c, d ∈ ℝ, ad - bc ≠ 0}} \bigr\}$, and H = $SL(2, ℝ) = \bigl\{ {{[\bigl(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\bigr)]: a, b, c, d ∈ ℝ, ad - bc = 1}} \bigr\}$. Then, ∀A ∈ G, the coset AH is the set of all 2 x 2 matrices with the same determinant as A.

Let d = det(A), d ≠ 0. If B ∈ SL(2, ℝ) ⇒ det(B) = 1 ⇒ det(AB) = det(A)det(B) = d·1 = d.

• Let G = U30 = {1, 7, 11, 13, 17, 19, 23, 29} and let H = {1, 11}.

1H = {1, 11}. 7H = {7, 7*11 mod 30} = {7, 17}
We don't need to calculate 11H or 17H, we already know that 1H = 11H and 7H = 17H. Next, we choose a representative not already appearing in the previously calculated cosets, say 13. 13H = {13, 23 (13*11 mod 30 = 143 mod 30 = 23)}.

19H = {19, 29}, so we have found the partitioning of G into equivalence classes under the equivalence relation defined by a ~Rb if and only if a-1b ∈ H or aH = bH, G/H = {1H, 7H, 13H, 19H} where 1H = 11H = {1, 11}, 7H = 17H = {7, 17}, 13H = 23H = {13, 23}, 19H = 29H = {19, 29}.

Definition. Let G be a group, and let H be a subgroup of G, H ≤ G. The set of cosets of H in G is denoted as G/H. It is called the quotient set of G by or mod H. G/H = {aH | a ∈ G}. The index of H in G is the number of distinct left (right) cosets of H in G. It is denoted as [G:H] = |G/H| where we say [G:H] = ∞ if G/H is infinite.

Example: k ∈ ℤ, k >0, kℤ is a subgroup of ℤ (kℤ ≤ ℤ). The quotient set is ℤ/kℤ. ℤ/kℤ = {[0], [1], … [k-1]} and [ℤ : kℤ] = k.

Theorem. Let H be a subgroup of G. Then, [G/H] = [H\G].

Proof.

Let Φ: G/H → H\G defined by Φ(gH) = Hg-1

• Φ is well-defined. Suppose gH = g’H, we claim Hg-1 = Hg’-1

Let x ∈ Hg-1 ⇒ ∃h ∈ H: x = hg-1 ⇒ x-1 = [Shoes and sock theorem] gh-1 ∈ gH = g’H ⇒ ∃k ∈ H: x-1 = g’k ⇒ x = [Shoes and socks theorem] k-1g’-1 ∈ Hg’-1 ⇒ Hg-1 ⊆ Hg’-1. Mutatis mutandis, that is, by a completely similar argument, Hg’-1 ⊆ Hg-1, hence Hg-1 = Hg’-1.

• Let ψ: H\G → G/H, ψ(Hg) = g-1H. By similar reasoning, this mapping is well-defined. It is indeed the inverse of Φ (Notice: ψ(Φ(gH)) = ψ(Hg-1) = (g-1)-1H = gH and Φ(ψ(Hg)) = Φ(g-1H) = H(g-1)-1 = Hg) ⇒ Φ is invertible and bijective ⇒ [G/H] = [H\G]∎

# Bibliography

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. This post relies heavily on the following resources, specially on NPTEL-NOC IITM, Introduction to Galois Theory, Michael Penn, and Contemporary Abstract Algebra, Joseph, A. Gallian.
1. NPTEL-NOC IITM, Introduction to Galois Theory.
2. Algebra, Second Edition, by Michael Artin.
3. LibreTexts, Abstract and Geometric Algebra, Abstract Algebra: Theory and Applications (Judson).
4. Field and Galois Theory, by Patrick Morandi. Springer.
5. Michael Penn (Abstract Algebra), and MathMajor.
6. Contemporary Abstract Algebra, Joseph, A. Gallian.
7. Andrew Misseldine: College Algebra and Abstract Algebra.
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