Abstract Alg, Lec 17B: External Direct Product Order Calculations, Isomorphisms, Normal Subgroups

Описание: The order of an element (a,b) in an external direct product G x H is the lcm of the orders of a and b, if they are finite order. Isomorphism facts about U(n) groups. Normal subgroup definition and examples. "Contemporary Abstract Algebra", by Joe Gallian: https://amzn.to/2ZqLc1J. Check out my blog at: https://infinityisreallybig.com/
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(0:00) Correct mistake from the end of Lecture 17A on the number of elements of order 4 in the external direct product of Z4 with itself.
(0:24) Example: Number of elements of various orders in Z2 + Z2 + Z4 (external direct product).
(4:43) Example: Number of elements of various orders in Z2 + Z2 + Z2 + Z2 (external direct product).
(5:40) Example: Number of elements of various orders in Z2 + Z8 (external direct product).
(8:05) All these examples are groups which are not isomorphic (and none are cyclic).
(9:06) Criterion for the external direct product of two finite cyclic groups to be cyclic and corollaries.
(11:28) The order of the groups in an external direct product is irrelevant (as far as isomorphism classes go).
(12:59) Isomorphism facts related to U-Groups (groups of units modulo n) and examples: U(70) and U(100)
(19:59) Introduction to normal subgroups, including the definition.
(22:47) Normality of a subgroup is not a transitive relation.
(23:55) Any subgroup of an Abelian group is a normal subgroup.
(24:55) The trivial subgroup, entire group, and center of a group are normal in the group (though the center might be trivial).
(26:28) In S3, the cyclic subgroup generated by the cycle (12) is not normal in S3.

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