Abstract Algebra Class 4: Group Definition, Examples, Cayley Tables, Isomorphic Groups

(0:00) Definition of a group. A nonempty set G is a group if there is a binary operation defined on G (so G is closed under the binary operation) such that (i) the binary operation is associative, (ii) there exists an identity element, and (iii) for each element of G, there is an inverse. Commutativity is not assumed because there are examples of algebraic structures that are not commutative (like matrix multiplication) and we would like them to be groups.
(12:12) Basic consequences of this definition include: (i) uniqueness of the identity, (ii) left and right cancellation laws, and (iii) uniqueness of inverses.
(16:20) Examples of groups and Cayley tables: (i) The trivial group (one element group), (ii) Z2 = {0,1} under addition mod 2, (iii) Z3 = {0, 1, 2} under addition mod 3,
(27:51) (iv) an abstract group of order 3 (introduce the idea of isomorphic groups here),
(34:57) (v) Z4 = {0, 1, 2, 3} under addition mod 4,
(38:14) (vi) U(8) = {1, 3, 5, 7} (the group of units under multiplication mod 8, as a set, this is the set of positive numbers less than 8 that are relatively prime to 8),
(44:45) (vii) G = {1, i, -1, -i} (every element is a power of i: this is called a cyclic group).
(53:33) Symmetries of an equilateral triangle (consisting of 3 rotations and 3 reflections) and its relationship to the symmetric group S3. Either way you think about it, the group elements are functions. The symmetric group S3 on 3 elements C3 = {1,2,3} consists of the 6 functions that permute these elements (these are one to one and onto functions from this set to itself). Compute the values of the 6 functions. This forms a group under function composition.

Abstract Algebra Screencast of Class 4 on February 8, 2021.

#abstractalgebra #grouptheory #groupdefinition

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[ericksonjohna]

Do elements listed in the leftmost column of a Cayley table get multiplied on the left, and elements listed int he topmost row get multiplied on the right?

[carterwoodson]

Looking forward to this! Thank you prof. Kinney!