%%visits: 3 Define an equivalence relation ~ by: - a b ⇔ ba−1 ∈ H
The equivalence class of a ∈ G under this equivalence relation:
[a] ≡ {b ∈ G|b a}, which is called right-coset S3 = ⟨a, b⟩ with a3 = e, b2 = e, b2 = e, ab = ba2
Cosets, are disjoint, same number of elements and they together cover the main set. Aka they are equipartition.
Two right cosets of G w.r.t H ⊂ G are either disjoint or identical.
Cosets: can be defined without group theory
Subsets : H ⊂ G with an equivalence_relation
Coset := Y ⊂ X with y1 y2∀y1, y2 ∈ Y
Suppose Y1, Y2 are both cosets of (X, ). then either y1H = y2H so they describe the same coset Or y2H ≠ y1H so they are not the same coset. Proof that all elements belong to a coset. :todo: ⊔yH = X cosets make partitions proof :todo:
What is a coset space?
The set of all cosets
Cosets are a sort of division, as they partition. ## rigour ## exam clinic when proving isomorphisms, think, are they the same elements, and do they have the same multiplication law. tags :math:
What is a right coset space? consider different coset spaces: [x] = [y] or [x] ∩ [y] = Φ
And one way to denote a coset is in element-subgroup notation.
H \ G
Coset space, what is the equivalence_relation that