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Unformatted text preview: Math 150a: Modern Algebra Cross-sections and complements Since the book does not much discuss cross-sections and complements, here are some notes that may be helpful. This topic is related to sections 2.8 and 2.10 of the book. If G is a group and A ⊆ G is a subgroup, then the left cosets { gA } of A are a partition of G . As a set, they are called G / A , the left coset space of A . A subset B ⊆ G is a cross-section of G / A if B has one element in each left coset of A . If B is also a subgroup of G , then it is a complement of A . More precisely, the kind of cross-section that I am describing is a left cross-section (because it cuts across left cosets), and if it is a subgroup, a left complement. However, a left complement is also a right complement (exercise GK4(a)). (A left cross-section doesn’t have to be a right cross-section.) There are five degrees of goodness of cross-sections and complements. 1. If A is not normal and B is not a subgroup, then G / A is not a group either. In this case you can still say that G / A ∼ = B as sets, and G ∼ = A × B as sets. This is a way to say thatas sets....
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BONUS NOTES - Math 150a Modern Algebra Cross-sections and...

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