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What's a Quotient Group, Really? Part 2

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Today we're resuming our informal chat on quotient groups. Previously we said that belonging to a (normal, say) subgroup N N of a group G G just means you satisfy some property. For example, 5 Z ⊂ Z 5Z⊂Z means "You belong to 5 Z 5Z if and only if you're divisible by 5". And the process of "taking the quotient" is the simple observation that every element in G G either #1) belongs to N or #2) doesn't belong to N and noting that the elements of G G can be grouped together according to HOW they satisfy either #1 or #2. The resulting 'piles' are precisely the cosets g N gN of G / N G/N . And the actual process of creating the piles is what the so-called "natural projection homomorphism" ϕ : G → G / N ϕ:G→G/N is doing when it sends an element g g to the coset g N gN . The representative g g simply indicates/describes how #1 or #2 is satisfied. Notice there is only one way to satisfy #1---you simply belong to N N ---but in general there can be ma

What's a Quotient Group, Really? Part 2 - math3ma    © 2015 – 2025 Math3ma Ps. 148 Archives November 22, 2016 • Algebra What's a Quotient Group, Really? Part 2 Today we're resuming our informal chat on quotient groups. Previously we said that belonging to a (normal, say) subgroup $N$ of a group $G$ just means you satisfy some property. For example, $5\mathbb{Z}\subset\mathbb{Z}$ means "You belong to $5\mathbb{Z}$ if and only if you're divisible by 5". And the process of "taking the quotient" is the simple observation that every element in $G$ either #1) belongs to N or #2) doesn't belong to

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