Abstract Alg, Lec 13: Visualize Cyclic Gps, S5 & A5, Isomorphisms & Properties, Intro Lagrange's Thm

Abstract Algebra, Lecture 13.

(0:00) Announcements.
(1:53) Lecture plan.
(2:46) Visualize finite cyclic groups on Mathematica as subgroups of S^1 (the unit circle group in the complex plane).
(10:54) The number of elements of various orders in S5 (symmetric group of degree 5) and A5 (alternating group (even permutations) of degree 5).
(18:31) Number of cyclic subgroups of various orders.
(22:27) Review definition of an isomorphism.
(24:56) Specific isomorphisms: 1) all cyclic groups of infinite order are isomorphic to Z (the integers under addition), 2) all cyclic groups of order n are isomorphic to Zn ({0,1,2,...,n-1} under addition mod n), 3) D3 (symmetries of an equilateral triangle) is isomorphic to S3 (symmetric group on 3 objects).
(34:04) Properties of isomorphisms acting on group elements.
(34:18) The identity gets mapped to the identity.
(37:49) Isomorphisms acting on powers of group elements.
(41:08) Isomorphisms acting on commuting elements.
(42:40) Isomorphisms acting on cyclic groups.
(46:25) Isomorphisms preserve orders.
(48:39) Preservation of number of solutions to equations and preservations of number of elements of various orders.
(49:55) Properties of isomorphisms acting on groups.
(50:20) The inverse function of an isomorphism is an isomorphism.
(56:40) G is Abelian iff its image (under an isomorphism) is Abelian & G is cyclic iff its image is cyclic.
(57:45) The image of a subgroup of G is a subgroup of the image of G (under an isomorphism).
(58:56) The preimage (inverse image) of a subgroup of the image is a subgroup of the original group.
(1:01:59) The image of the center of a group is the center of the image of the group (under an isomorphism).
(1:02:40) Reminder to read about Aut(G) and Inn(G) (the group of automorphisms of G and the group of inner automorphisms of G).
(1:03:10) Lagrange's Theorem (Fundamental Theorem of (Finite) Group Theory) and Corollaries.

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[sherryj]

Hello sir,
Thanks for your course .Please if you can help me with this-
What is the highest and second highest order of an element in S10 ?Also, find the number of elements with highest order ?
My answer is-a)For highest it should be disjoint cycles of lengths 5, 3,2 so order should be 30 .And no of elements should be 10!/5*3*2.
b)For second highest it should be disjoint cycles of 7 and 3, so order should be 21.

I am not sure about my answers.Please, if you can verify if they are correct.If they are wrong, please help me with the solution.
Thanks for your help .

[andrejcermak]

56:40
could this proof be done by saying g1 bar = phi(g1) and using phi^-1(phi(x)) = x?