Abstract Algebra, Lec 18B: Normal Subgroups, Factor Groups (a.k.a. Quotient Groups)

(0:00) Mention typo from previous slide.
(0:45) Example of a normal subgroup: H = A3 is normal in G = S3.
(7:09) The normal subgroup test and an example: outline of proof that {(1), (12)(34), (13)(24), (14)(23)} is normal in A4.
(14:54) Comment about when HK is a subgroup of G.
(15:32) Definition of a factor (quotient) group, both as a set, and with its binary operation of coset multiplication (aH)(bH) = abH (and in additive notation: (a + H) + (b + H) = (a + b) + H.
(19:15) Quickly mention that the group operation of coset multiplication must be shown to be well-defined.
(19:40) Example: Z/H, where H is the cyclic subgroup of Z generated by 3. This is isomorphic to Z3 (make Cayley table).
(25:53) The G/Z Theorem: If G/Z(G) is cyclic, then G is Abelian.

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[shotpana]

Hey professor! thanks again for the videos. I'm just wondering if there is a lecture video on "isomorphism"... I couldn't find one. Thanks again.

[余淼-eb]

Hello professor, you have shown that A_3 and A_4 are normal subgroups of S_3 and S_4 respectively. How to prove that A_n is a normal subgroup of S_n. Thanks so much.

[harshvardhansingh]

Professor, What is the definition of product of Hx and Hy ? Is (Hx)(Hy) is defined to be equal to H(x.y) ? Ideally, (Hx)(Hy) should be the set obtained when all the elements of Hx are done binary operation with all the elements of Hy . Can we prove that when we do (Hx).(Hy), it comes out to be equal to Hxy ?

[sherryj]

Hello Sir,
At 10:55 you have mentioned that any subgroup of order 4 is isomorphic to either Z4 or Z2 X Z2. Is this some theorem or standard result ?As far as i can recall, we have not proved it in this lecture series .Please, if you can help me with this prove .
Thanks a lot for the videos.