Abstract Algebra, Lec 9A: U(n) Group Cayley Table on Mathematica, Proof of Theorem on Cyclic Groups

(0:00) Reminder about Exam 1.
(1:00) Lecture overview.
(1:34) Mathematica Code to Make Cayley Tables for U(n), the group of units under multiplication modulo n.
(5:46) Criterion for a^i = a^j (both when "a" has infinite order and finite order.
(7:38) Proof of the criterion when the order |a| = infinity.
(12:11) Proof of the criterion (set equality portion) when the order |a| = n (note r should be less than n in proof).
(20:10) Proof of the criterion for a^i = a^j when the order |a| = n.
(25:54) Corollaries of condition for when a^i = a^j.
(30:42) Orders of (and subgroups generated by) powers of generators of finite cyclic groups.
(31:44) Lemma: Suppose G is a group, H is a subgroup of G, and "a" is an element of G. If a is an element of H, then the cyclic subgroup of G generated by "a" is a subgroup of H (and its proof).

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[ritesharora]

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[paulmyrin]

Would you please provide access to your previous homework assignments and practice tests?