Abstract Algebra, Lec 8B: Cyclic Groups: Empirical Observation of Properties

(0:00) Criterion for a^i = a^j. Example: cyclic group of order 8. If |a| = n, then a^i = a^j iff n divides i - j.
(5:16) Corollaries: the order of an element and the order of the cyclic subgroup generated by the element are the same; if |a| = n and a^k = e, then n divides k; |ab| divides |a|*|b| when a and b commute;
(11:28) Theorem: If |a| = n, then |a^k| = n/gcd(n,k), arrived at by considering the example with |a| = 8.
(16:44) If |a| = n and k and n are relatively prime, then a^k is a generator of the cyclic subgroup generated by "a"; also think about this when |a| = 8 and relate to elements of U(8), the group of units modulo 8 under multiplication.
(19:37) Corollaries: order of an element divides the order of the group; if |a| = n, then |a^i| = |a^j| iff gcd(n,i) = gcd(n,j); generators of finite cyclic groups.
(22:13) Fundamental Theorem of Cyclic Groups: subgroups of cyclic groups are cyclic; the order of any subgroup divides the order of the group; there is exactly one subgroup for each order that divides the order of the group; mention subgroup lattice;
(24:53) Euler phi function: make a table of values, state relationship to the number of generators of a cyclic group of order n, and then plot it on Mathematica with DiscretePlot.

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[andrejcermak]

Hi! I was just curious, but whats the content of those quizes? Are they marked? Thank you!