Quotient Groups and Homomorphic Images | Abstract Algebra

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We introduce quotient groups/factor groups. The quotient group of a group G by a normal subgroup H is the set of all cosets of H in G with the operation of coset multiplication. We prove the quotient group construction is indeed a group, and that it is a homomorphic image of the containing group. Thus, quotient groups give us a way to construct homomorphic images of a group. #abstractalgebra #grouptheory

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Here's what I didn't understand at first, about 9:40 you talk about adding the coset <5>+1 to <5>+3 and saying it would equal the coset <5>+4, but if you add the elements shown, you get (-16, -6, 4, 14, 24). So you're missing the -1 and 9 which are elements of <5>+4. But you'd be adding every element in <5>+1 to <5>+3, and that would generate the -1 and 9, which is kind of an insane notion (adding infinitely many integers to infinitely integers to get a list of infinite integers)

MrCoreyTexas

What exactly is a homomorphic image? And f:G —> G/H is a surjective but non injective mapping right? Im thinking about Z to Z/5z. Z/5z has 5 elements. 1, 6, 11, 16…. are mapped to 1 and so on.

cuneytkaymak

Quotient group is set of all cosets. But wouldnt the set of cosets be the group itself? Since all cosets of a subgroup are disjoint and all of them will contain all of the elements of group itself....?

archiem

Where were these videos when I was struggling with Abstract Where was Gondor when my GPA fell?

(Thanks tho, because I still need to understand it for set theory apparently. Maybe I can learn to not hate it.)

shambo

can you do the video on examples of quotient groups? thanks

James-eowh

Excellent lecture. Nice hair style, BTW. 😀

wenzhang

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