Chiusakpung's Electronics Blog

평소 Group Theory에 극히 목말랐던 저는 오늘 친구한테 Fraleigh가 쓴 A First Course in Abstract Algebra를 빌렸습니다. W.S. Yoon, D.G. Lee, Thank you very much! (이들이 쓴 수학 이론이 세계를 놀라게할 날이 오기를!)

그리고 혹시나 해서 Youtube에 검색했더니 아래의 동영상들 발견 ㅋ
-_- 참 재미있는 양반입니다. 이분 덕에 Group Theory가 더욱 더 재미있게 느껴집니다.

. .

헉 *_* 너무 재미있어서 2시간은 가뿐히 봤네요. 이 동영상 덕분에 Modern Algebra 1의 중간고사 범위까지는 개념적이나마 쓱~ 훑어볼 수 있었습니다!

Basic abstract algebra, pt.1 - Sets, Functions

Basic abstract algebra, pt.2 - Binary Operator, Group(Closure, Assoc., Identity, Inverse)

Basic abstract algebra, pt.3 - Examples of Groups, Thm1
If G is a group, there is only one element e

Basic abstract algebra, pt.4 - Operation Table(See how he construct it!)
Does a+x=b on even integers always have a solution? Check its algebraic structure(Group!)

Basic abstract algebra, pt.5 - Surjective, Injective, Bijective, Homomorphism, Isomorphism(Isomorphic)

Basic abstract algebra, pt.6 - Examples of Homomorphism & Isomorphism
check 1. f(a*b) = f(a) # f(b) 2. It is injective 3. It is surjective
Notation : Let nZ be the group consisiting of multiples of n with addition

Basic abstract algebra, pt.7 - Abstract Algebra

Basic abstract algebra, pt.8 - abelian group, subgroups
if 2 groups are isomorphic, it doesn't mean that the groups are equal
But! if two groups are isomorphic, they are algebraically equivalent - or structurally the same group
if 2 groups are isomorphic, both are abelian or both are non-abelian
Let G be a group w/ binary operator *
A subset H of G is a subgroup of G if H is a also a group under the binary operator *
A subgroup can in fact be isomorphic to the group it is a subgroup of!! (Think nZ & Z)
= ∃an injective homomorphism from H to G
also, the identity for G = the identity for H

Basic abstract algebra, pt.9 - Notations for addition & multiplication
Def1. H is a subgroup of G if ∀a, b ∈ H → ab^-1 ∈ H
Def2. Let G be a group & S ⊆ G.  <s> is the smallest subgroup of G that contains S.
<s> is called the subgroup of G generated by S
Def3. If a group G can be generated by a finite set, we say the G is finitely generated.
If G is a group generated by only one element, we say that G is cyclic.
Ex. = nZ of Z. is the cyclic group generated by n.
All cyclic groups which are infinite, are isomorphic to Z
Thm1. A cyclic group is abelian (6:43)

Basic abstract algebra, pt.10 - Number Theory
n = mq + r, 0 ≤ r < |m| for integers
if r = 0, m divieds n, m | n
Def. a, b are congruent modulo n if n | (a-b), a ≡ b (mod n)

Basic abstract algebra, pt.11 - More about Modulo -> Group
n possible remainders : 0 ~ n-1

Basic abstract algebra, pt.12 - Z_n={0, 1,2...(n-1)}
where k = {a| a≡k(mod n) }
So we can set up a group on Z _n (which is the set of sets!)