Abstract Algebra, Lec 7B: Intersection of Subgroups is a Subgroup, Cyclic Groups & Non-Cyclic Groups

Abstract Algebra, Lecture 7B.

(0:00) Review the end of Lecture 7A: the center Z(G) of a group G equals the intersection of centralizers of all the elements.
(0:27) The intersection of any collection of subgroups of G is a subgroup of G. Give the idea of the proof.
(4:01) Subgroup generated by a set S of a group G.
(10:15) Examples of cyclic groups and generators, as well as non-cyclic groups (which don't have generators) (also review what a cyclic group is).
(21:33) Questions and empirical observations about cyclic groups: Is the number of generators of a cyclic group determined by the order of the cyclic group? (Do calculations in an abstract cyclic group of order 4).
(29:27) Can we determine whether two powers of a generator result in equal elements or not? If |a| = 5, is a^73 = a^32 (the answer is no).
(31:41) If we have a generator, can we find the orders of its powers? How do the orders of the elements of a cyclic group relate to the order of the group? What types of subgroups do cyclic groups have?

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[harshvardhansingh]

Sir, at 23:21, I am unable to understand why a multiplied 4 times by itself gives e. Why the order of an element is equal to the order of its cyclic subgroup generated by the element ?

[harshvardhansingh]

@Bill Kinney, Sir at 6:59, you said G is group of 'positive integers' under addition . When we check the smallest subgroup of G that contains <6, 8>, we find that it contains negative integers too. Shouldn't G be then called group of 'integers' under addition ?