Abstract Alg, Lec 12A: Fund Thm Cyclic Groups, Groups of Order 1, 2, 3, 4, or prime, Symmetric Gp S4

(0:00) Lecture plan.
(0:17) For each divisor k of a cyclic group of order n, there is exactly one cyclic subgroup of order k (part of the Fundamental Theorem of Cyclic Groups).
(4:23) Any group of order 1 is cyclic.
(5:12) Any group of order 2 is cyclic.
(7:20) Any group of order 3 is cyclic.
(15:25) A group of order 4 does not have to be cyclic: for example, U(8) is not cyclic (Z4 and U(10) on the other hand, are cyclic).
(19:08) The quality of being isomorphic is an equivalence relation on the collection of all groups.
(25:03) Subgroups of S4 (Symmetric Group on 4 objects, a.k.a. Symmetric Group of Degree 4), by "Type".

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[Sipnayan]

Hi Professor Kinney. I'm learning Abstract Algebra again, on my own, and believe it or not, I'm following every video in this course and taking down notes as if I'm in your class. Again, thank you very much for sharing the videos.

After this, I plan to "take" the Real Analysis course.

[aashishdubey]

Sir... I m very much affected by ur teaching sense... Can u please help me to understand this theorem?

[aashishdubey]

if G is a finite group of order n and for every divisor d of n, there exists a unique subgroup of order d then G is cyclic.