Normal Subgroups and Quotient Groups -- Abstract Algebra 11

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[schweinmachtbree]

There is a problem with left vs right cosets in this video - e.g. the equality of left cosets aH = cH is equivalent to the condition a^(-1)c ϵ H, not ca^(-1) ϵ H. So the argument on the board at 10:00 becomes correct when all the left cosets are changed to right cosets, where ca^(-1) ϵ H etc. is the correct condition - that is, change gH to Hg throughout.

[matematicacommarcospaulo]

I liked very much the way you introduce normal subgroups in a constructive way... We don't see this approach, for instance, in the classic book of Herstein

[noahtaul]

32:27 You’re not actually working with the quotient here, you’re working with the subgroup.

37:24 Same thing, your symmetry group is now <r^2>, nothing to do with the quotient. To talk about the quotient, you need to describe characteristics of the square that are unchanged under the subgroup action. For instance, with the subgroup <r>, the clockwise order of the sides is unchanged, and for the subgroup <r^2, s>, the pair of parallel edges on top and bottom is unchanged, so the quotient groups act on those things instead of the square itself.

[bjornfeuerbacher]

42:00 Wouldn't it be more sensible to point out that this quotient group is isomorphic to D_2?

[AbuMaxime]

Thank you for making me more intelligent when the rest of the world tries to make me more dumb.

[jordimartinez]

Are we going to see some Galois theory?

[karl]

Sorry, I'm not sure, ist THIS valid: In your "not-well-defined" example, the representations on the left have a 2-cycle as representative, so they should be elements of order 2 in the "new" group, but on the right they are represented by 3-cycles, so they should be of order 3 - contradiction!

[M.athematech]

The exposition of the proof of well-definedness of the multiplication of cosets is not correct because you make use of the operation in the proof before it is actually shown to be well defined.

[psychSage]

ваши размышления напомнили мне забавное высказывание: "математика - искусство называть разными именами одни вещи"