Abstract Algebra, Lec 4A: Group Definition, Basic Properties of Groups, Cayley Tables, Observations
What is a group in abstract algebra? Find out the definition in this lecture. "Contemporary Abstract Algebra", by Joe Gallian: https://amzn.to/2ZqLc1J. Check out my blog at: https://infinityisreallybig.com/. Abstract Algebra Playlist: • Abstract Algebra Course, Lecture 1: Introd...
Abstract Algebra Course, Lecture 4A.
(0:00) Super Bowl comments.
(0:35) Have a clicker for powerpoint and will use overhead projector today.
(1:06) Lecture summary.
(1:35) Definition of a group: binary operation (closure), associativity, existence of identity, existence of inverses.
(8:06) Prove that the identity element is unique.
(13:24) Prove the left cancellation law.
(20:38) Cayley tables for Z3 and Z4.
(27:45) Introduction to the idea of a subgroup (like a subspace of a vector space from Linear Algebra).
(29:37) Review U(n) (the group of units modulo n, under multiplication mod n).
(31:02) Make Cayley tables for U(8) and U(10).
(38:31) Observation that U(8) and U(10) are not isomorphic.
Bill Kinney, Bethel University mathematics department
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