Abstract Algebra, Lec 14B: Automorphisms, Inner Automorphisms, Lagrange's Theorem, and Cosets

(0:00) Automorphisms and inner automorphisms.
(2:00) Verification that any inner automorphism is one-to-one, onto, and operation-preserving.
(5:59) Elements of the center of a group generate the identity inner automorphism.
(7:18) Aut(G) and Inn(G) are groups under function composition. Also note that Inn(G) is trivial if G is Abelian.
(9:12) Aut(Zn) is isomorphic to U(n) and the idea of the proof.
(15:40) Lagrange's Theorem (Fundamental Theorem of Finite Group Theory) and corollaries.
(18:55) Special notation related to cosets.
(20:41) Left and right cosets containing an element "a" (a representative of the coset).
(21:53) Example: consider some left and right cosets of H = the subgroup generated by the element V in D3.
(26:45) Properties of cosets.

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[ldb]

I understand everything in the lecture except for one thing. How is there a light switch built into the white board?

[WahranRai]

20:45 what is the domain of a in the coset definition ?
G group, H subset of G, h belongs to H and a belongs to ?

[sherryj]

Hello Sir,
At 12:30 you have mentioned that alpha(k)=alpha (1+1+1....k times) and then you have applied property of isomorphism to say that it is equal to k times alpha( 1).But, here Zn is a group under addition modulo n .Then why are you taking the operation as addition?I think it should be addition modulo n.Please help me with this.
Thanks