Abstract Algebra, Lec 4A: Group Definition, Basic Properties of Groups, Cayley Tables, Observations

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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[andrejcermak]

great lectures so far! due to the corona virus, we probably wont have any lectures this semester, so we have to study at home.

[ll]

When a group of four elements is defined i.e. Z4, I think this notation has something wrong. I should be able to define my group using a notation that belongs entirely to the group. It’s like I need something that doesn’t exist to describe what actually exists, my group and nothing more. I would describe Z4 as a group of 3 element plus the neutral element e.

[shiptoaster]

What does the "=" relation actually mean algebraically? In set theory I understand that "=" is the conjunction of two facts about the subset relation, which has a clear and unambiguous definition. A is a subset of B if every element of A is an element of B--set membership seems somehow fundamental or self-evident as a property in a way that algebraic equality does not.

If I write 3+1=4 in Z, it doesn't seem obvious to me that these expressions represent the "same" thing, one is just an element of the set and one is the output of a function on the set. Just as in, not as tautological as it would be to say 4=4, there is some minimal amount of meaning encoded in 3+1=4.

Similarly in a group G in general ab=ac means a particular function, the binary operation of the group, takes on the same value at different points in the set GxG, (a, b) and (a, c). But what does b=c mean? The tautology of 4=4 is gone because the representations of the left and right side are literally different. Now it seems to encode some meaning. Can I interpret b = {b} and c = {c} so that b subset c and c subset b? That seems wrong somehow.

[adaelasm]

Is anyone else put off that in the left cancellation law it is taken for granted that b=c => a*b=a*c? It doesn't seem obvious why that this something we can assume without proof but we need to prove the other direction.

[nikolaskoutroulakis]

Isn't the proof that "if b=c then ab=ac" trivial? The lecturer mentioned he needed this to prove the converse holds, but here is a simple way to prove it.

Suppose b = c. It is necessarily true that ab = ab. Now use identity substitution to get ab=ac

[tugbadnl]

Teacher, Thaank you very much.from turkey.Please turkish translate

[ShwetankT]

Do you intend to do Galois theory in this lecture series? How many lectures are you able to take in a semester? #amazed

[handle]

for the left cancellation law proof/segment, we are assuming a!=0? I appreciate your response