Mont Gum Did you know you can slow down the video playback? Click on the little "Gear" icon and try Speed 0.5. I sound a little funny but the pace is easier to keep up with :)
It has to be fast. This video covers a 50 minute lecture in 11 minutes 36 seconds. If you find this video ‘fast,’ please consider... 1. Watch this video multiple times. Its structure, incomprehensible in the first viewing, becomes comprehensible in subsequent viewings (Gestalt). 2. Do a quotient groups search: www.google.com/search?q=quotient+groups Recommend Millersville: www.millersville.edu/~bikenaga/abstract-algebra-1/normal/normal.html Also recommend its Abstract Algebra 1 series: www.millersville.edu/~bikenaga/abstract-algebra-1/abstract-algebra-1-notes.html
i like your videos a lot, but the speed at which you talk is so fast that i can't keep up. but still you're my favorite so far. just wish i wasn't having a panic attack the whole time.
In the "Quotient Mapping Theorem", you refer to an epimorphism from G to G/N, but since we aren't necessarily assuming that G/N is a group (since N is not necessarily assumed to be normal), it seems like we should not be using the word "epimorphism".
So at first I figured that quotient groups are very similar to groups but different in that the elements are not sets but cosets (I also originally wondered why they aren't called cosubgroups haha). Now if my interpretation is correct I see the quotient group to still be a set of elements namely for Z mod 12 { {0,3,6,9},{1,4,7,10},{2,5,8,11} }. In set theory these sets (the cosets) are valid elements of a set. So we strip the subgroup of its binary op and use it and the other cosets for the set.
Joanne Koratich Thanks! And you're right, taking (1,1) as a generator in Z2xZ3 shows that that group is in fact isomorphic to Z6, so they shouldn't be separate (except insofar as to make this point).
5 years late, but perhaps for other viewers: they're both common functions in AA1 (since this is AA2). The image of a function is the codomain. The kernel of a function consists of all elements in the domain that are sent to the identity element in the codomain.
Okay, I think I get it :) Thanks Professor. And on another note I think you should find I updated the notes pdf in dropbox. I am not used to dropbox, so I hope I did that correctly.
Hello Professor Salomone, would you mind providing some references for the proof of the Fourth Isomorphism Theorem? It seems that this proof can not be found in Gallian's book. Is there any other text book recommended? Thanks so much.
" the kernel of any mapping out of G is isomorphic to a normal subgroup of G " Better: the kernel of any homomorphism out of G *is* a normal subgroup of G, not merely up to isomorphism.
Actually I think I understand this! So since there is a kernel to the mapping of a homormorphism out of G that is contained in every codomain group and every kernel is a normal subgroup, the kernel of any mapping out of G is isomorphic to a normal subgroup of G okay okay I got it! That is the 1-1 correspondence part Wow so any homomorphism you want and you have associated a normal subgroup in your codomain..quite powerful
