Cynosure对《Abstract Algebra(Second Edition)》的笔记(1)

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数学归纳法有2种形式。

a | b: a divides b. b被a整除, e.g. 2 | 6
a ∤ b

proper subset
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Section 1 BINARY OPERATIONS

- the abstract approach may clarify our thinking about familiar situations by stripping away irrelevant aspects of what is happening.
- it may lead us to consider new systems that are valuable because they shed light on old problems. 
- a general approach can save us effort by dealing with a number of specific situations all at once. 
- although it can take some time to appreciate this, abstraction can be just plain beautiful.
 DEFINITION   Suppose that:
axioms i)G is a set and * is a binary operation on G,
axioms ii)* is associative,
axioms iii)there is an element e in G such that x*e = e*x = x for all x in G, and
axioms iv)for each element x∈G there is an element y∈G such that x*y = y*x =e.
Then G, together with the binary operation *, is called a #group#, and is denoted by (G, *).


Group + commutative :  #abelian group#


AΔB:symmetric difference

P(X): power set of X

GL(2, R) : general linear group of degree 2 over R

a̅  : remainder of a mod n.

two integers a 1 and a 2 have the same remainder mod n iff a 1 - a 2 is a multiple of n. we say that a1 and a2 are congruent modulo n, and we write a1≡a2 (mod n).
.
x⊕y ≡(x+y) ̅
The group (Zn, ⊕) is called the #additive group of integers mod n#. Notice that, for n = 1, we have Z1 = {0}, so (Z1, ⊕) is a group with only one element in it.  
In general, any group having only one element is called trivial.  If x is any object whatsoever (e.g., x =Whistler's Mother), then we get a trivial group ({x},*) by defining x*x=x.
.
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SECTION3  FUNDAMENTAL THEOREMS  ABOUT GROUPS

First, a convention: We usually call the operation in a group "multiplication"; but very often the operation is called "addition" if the group happens to be abelian.

another formulation of the axioms for group:
Let G be a set and * an associative_ binary operation on G. Assume that there is an element e∈G such that x*e=x for all x∈G, and assume that for any x∈G there exists an element y∈G such that x *y = e.  Then (G, *) is a group.

An analogous proof shows that assuming associativity and the existence
of a left identity and left inverses is also sufficient to guarantee a group. It
should be observed, however, that associativity plus the existence of a right
identity and left inverses (or a left identity and right inverses) is not enough.

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SECTION 4 POWERS OF AN ELEMENT : CYCLIC GROUPS

DEFINITIONS If G is a group and x∈G, then x is said to be of finite order if
there exists a positive integer n such that x^n =e. If such an integer exists, then
the smallest positive n such that x^n = e is called the #order of x# and denoted by
o(x). If xis not of finite order, then we say that xis of infinite order and write
o(x)= ∞.

(m,n): the greatest common divisor (g.c.d.) of m and n. (m,n are integers, not both zero)
.
Euclidean algorithm:   if m=qn+r,  then (m,n)=(n,r)
.
THEOREM 4.2 
If m and n are integers, not both zero, then there exist integers x and y such that
mx+ny=(m,n)    （ (m,n)是m和n的整数线性表示）
Thus the g.c.d. of m and n can be written as a "linear combination" of m and n, with integer coefficients.
.
[m,n]: least common multiple (abbrev.   l.c.m)
习题4.29，4.31是g.c.d和l.c.m的类比
。
TIIEOREM 4.4
Let G be a group and x E G.
i) o(x)=o(x^-1).
ii) If o(x)=n and x^m=e, then n divides m.
iii) If o(x)=n and (m,n)=d, then o(x^m)=n/d.
.
A central goal of group theory is to classify all groups, i.e., to see what kinds of groups there are.
.
A group G is called #cyclic# if there is an element x∈G such that G= {x^n| n∈Z}; x is then called a #generator# for G.
 <x> : cyclic group.   x is the generator. G是由x(对于算符*)的powers组成的。

e.g. in additive notation, <x> = {nx| n∈G}.
.
THEOREM 4.5 Let G=<x>. 
If o(x)=∞, then x^j !=x^k for j !=k, and consequently G is infinite. 
If o(x)= n, then x^j = x^k iff j = k (mod n), and consequently the distinct elements of G are e, x, x^2 , ••• ,x^(n-l)
.
|G|：the order of  G，the number of elements in G
If G=<x>, then |G| = o(x)
cyclic group is alelian group
(反之alelian group 不一定是cyclic group，例如Klein 4-group)
The German word for "4-group" is "Viergruppe," and the 4-group is often denoted by V
.
Let G be a group and let a ∈ G. An element b∈G is called a #conjugate# of a if
there exists an element x∈G such that b = xax^-1. Show that any conjugate of
a has the same order as a.
H and K are conjugate if K=gHg^-1 for some g∈G
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SECTION 5 SUBGROUPS
A subset H of a group ( G,*) is called a #subgroup# of G if the elements of H form a group under *
.
THEOREM 5.1 (如何判断G的子集H是否是子群) Let H be a nonempty subset of a group G. Then H is a subgroup of G if and only if the following two conditions are satisfied:
i) for all a,b∈H, ab ∈H, and    (H is closed)
ii) for all a∈H, a^-1 ∈H            (H is closed under inverses)
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A subgroup H of G is #proper# if H != G
。
THEOREM 5.2     Let G be a cyclic group. Then every subgroup of G is cyclic
。
It is often true in mathematics that the proof of a theorem contains substantially more information than the statement of the theorem itself.
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Z( G)= {z ∈ G | zx= xz for all x ∈ G}： the center of G
center is always a subgroup
。
THEOREM 5.3 (简化子群的判断条件)
Let G be a group and let H be a finite nonempty subset of G. Then if  H is closed under  multiplication in G, H is a subgroup of G.
。
Q8：group of unit quaternions
normal subgroups
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THEOREM 5.4 (G的子集H和K的集合运算结果是否是子群)
Let H and K be subgroups of a group G. Then:
i) H∩K is always a subgroup of G; and
ii) H ∪K is a subgroup if and only if one of H, K is contained in the other.
。
a cyclic group of order n has exactly τ(n) subgroups。
τ(n) ： the number of positive divisors of  n
。
COROLLARY 5.6 (如何生成cyclic group的所有的subgroup)
THEOREM 5.7 (如何生成infinite cyclic group的所有的subgroup)
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SECTION  6    DIRECT PRODUCTS
前面的内容讲的是如何在G里选择一些元素得到一个新的群H。（称H是G的subgroup）。
本章讲的是，如何把一些group "patch together"，得到新的group。这有很多方法，最简单、常用的是
direct product。
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the direct product of G and H: GxH ≡ (g1g2, h1h2) = (g1, h1)(g2, h2),
G and H are called factors of GxH
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If A is a subgroup of G, B is a subgroup of H, then AxB is a subgroup of GxH
.
(在什么情况下，直积得到的group是cyclic group) 
THEOREM 6.1      Let G = G1 x G2 x · · · x Gn.
i) If gi∈Gi for 1<=i<=n, and each gi has finite order, then o((g1,g2, ... ,gn)) is the least common multiple of o(g1), o(g2), ... , o(gn).
ii) If each Gi is a cyclic group of finite order, then G is cyclic iff  |Gi| and |Gj| are relatively prime for i != j
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SECTION  7  FUNCTIONS

DEFINITION 
If S and T are sets, then a #function# f from S to T assigns to each s∈S a unique element f(s)∈T。

As a definition this is somewhat strange, in that it tells you what a
function does rather than what it is. Sometimes this difficulty is avoided by
saying that a function is a "rule" that assigns elements of T to the elements of
S, but this isn't any better because "rule" isn't defined. Besides, for some
functions the "rule" is obscure at best, and it may be so hard to state that
most people wouldn't call it a rule at all.
The above definition is fine as a working definition, and is how we
usually think of functions. A more precise definition is as follows.

DEFINITION (precise) 
A #function# from S to T is a set of ordered pairs (s, t), where each s∈S and each t∈ T, such that each s∈ S occurs as the first element of one and only one pair (s,t).

f: S->T

Let f: S->T be a function. f is #onto# if for each t ∈T there is at least one s∈S such that f(s)= t.
f is #one-to-one# if whenever s  and s2 are two different elements in S, we have f(s1)!=f(s2).

Onto(surjective): 
everything in T gets hit by f.  f is onto just if it doesn't miss anything in T
one-to-one (injective):
f  is one-to-one just if nothing in T gets hit twice. That is, anything in T is hit either just once or not at all.

S: Domain of f
T: Image of f
.
let f: S->T and g : T->U be functions, then #composite function#  gof: S->U is (gof)(s)≡g(f(s))
.
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SECTIONS  8  SYMMETRIC GROUPS
If X is a nonempty set, then a one-to-one onto mapping X ->X is called a #permutation# of X. 
the set of all such permutations forms a group (Sx, o) under composition of functions.
(Sx, o) is called the #symmetric group# on X
.
every group can be thought of as a subgroup of some  symmetric group.
.
如果X有n个元素，则Sx称为symmetric group of degree n，简记为Sn
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cycles ⊂ permutation
.
disjoint没看明白，参考《Contemporary Abstract Algebra》P103.
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transposition: a cycle that just interchanges two elements
.
n! (read "n factorial")
.
A($_n$) denote the subset of Sn consisting of all the even permutations
A($_n$) is called the #alternating group# of degree n
.
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About   S3
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S3 has 6 elements: e, (1,2, 3), (1,3,2), (2,3), (1,2), (1,3),
f ≡ (1,2, 3),    so f^2 = (1, 3, 2),    f^3 = e, <f> = {e,  f,  f^2}
g≡(2,3),          so g^2 = e,                              <g> = {e, g}
(如果把S3看成等边三角形ABC的话，f相当于绕重心旋转120度，g相当于B到C的折叠)
f^2g = gf
gf^2 = fg    (便于记忆：g f^i = f^-i g,   i=1, 2)
(fg)(fg) = e,   (f^2g)(f^2g) = e
S3 = { e,  f,  f^2,  g,  fg,  f^2g }
cyclic subgroups <e>, <f>, <f^2>, <g>, <fg>,  <f^2g>.
S 3 is that it is nonabelian, (because fg != gf)
.
D4 denotes the subset of S 4 consisting of the eight permutations that come from the symmetries of P4
D4 is a subgroup of S4 . 
D4 is called the #octic group#, or #group of symmetries of a square#
f≡(1,2,3,4), （旋转90度）
g≡(2, 4) （沿对角线折叠顶点2和4）
<f> = {e, f, f^2, f^3}
D4 = { e,    f, f^2, f^3,   g,   fg, f^2g, f^3g}
g^2 = e
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
gfg = f^-1
e: Place1=A, Place2=B, Place3=C, place4=D
g: Place1=A, Place2=D, Place3=C, place4=B
fg: Place1=B, Place2=A, Place3=D, place4=C
gfg: Place1=B, Place2=C, Place3=D, place4=A （注意，这里是把place2和place4位置上的值flip）
f^-1: Place1=B, Place2=C, Place3=D, place4=A
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
(gfg)^i = f^-i   ==>  gf^i g = f^-i  ==> gf^i = f^-i g   (just as for S3),
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SECTION9   EQUIVALENCE RELATIONS;   COSETS
what do we mean by a "relationship" that may or may not hold between two elements of S?
By a #relation# R on S we mean @a set of ordered pairs@ of elements of S. If s1 ,s2 ∈ S, then s1 is in the relationship R to s2  iff  the ordered pair (s1, s2) ∈ R.
把relation R定义为一个“有序对的集合”。该集合里的元素(s1,s2)表明s1和s2满足关系R，即s1 R s2

DEFINITION 
A relation R on S is called an #equivalence# relation on S if R has the following three properties:
Reflexivity:      For every s∈S, sRs;
Symmetry:       For every s1 and s2 in S, if   s1 R s2,   then   s2 R s1
Transitivity:    For every s1, s2, s3 in S, if   s1 R s2,  s2 R s3,   then s1 R s3
(要让定义的“相等关系”和我们的经验相符，那么这个定义必须满足上面三个性质)
。
s_={x∈S | xRs}, s_ is called the #equivalence class of S under R#(在S里满足R的元素构成一个equivalence class )
.
The elements of an equivalence class are sometimes called #representatives# of that class.
。
x, y∈ group G, and x,y 不同时属于子群H。这样x,y之间就存在了“差异”。可以来描述这种差异：
x ≡H y   iff   x y^-1 ∈H.
.
THEOREM 9.2 For any group G and any subgroup H of G, the relation ≡H defined above is an equivalence relation.
.
DEFINITION 
If H is a subgroup of G, then by a #right coset# of H in G we mean a subset of G of the form Ha, where a∈G and Ha={ha | h∈H}.
Ha:  the right coset of H in G.      Ha= { ha | H is a subgroup of G,  ∀h∈H , a∈G }(这么写正确吗？)
Ha is a "translate" of H by  element  'a'(把H的所有元素 "translate" ‘a'，得到Ha)
.
THEOREM 9.3 Let H be a subgroup of G. For a∈ G, let a_ denote the equivalence class of a under ≡H. Then: a_ = Ha.    
It means the equivalence classes of ≡H are precisely the right cosets of H.
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SECTION  l0  COUNTING THE ELEMENTS   OF A FINITE GROUP 

TIIEOREM 10.1 (Lagrange's Theorem) subgroup of G. Then Let G be a finite group and let H be a
|H| divides |G|
|G| = [G:H] * |H|
[G:H] is the #index# of H in G


LEMMA 10.2     Let G be any group (not necessarily finite), and let H be a subgroup of G. Let Ha and Hb be right cosets of H in G. Then there is a one-to-one correspondence between the elements of Ha and those of Hb.
==>  any two right cosets of H in G have the same cardinality.
。
如果集合S和T存在一一影射，则说S和T有相同的cardinality, 记为|S| = |T|
. 
TIIEOREM 10.3   Let H be a subgroup of G. Then the number of left cosets of H in G is [G:H]
THEOREM 10.4   Let G be a finite group and let x∈G. Then o(x) divides |G|. Consequently, x^|G| = e for every x∈G.
THEOREM 10.5   Let G be a group and assume that |G| is a prime. Then G is  cyclic. Moreover, any element of G other than e is a generator for G.
。
m⊙n ≡ (mn) ̅. where  ̅ denotes remainders modulo p.
.
Z(y): the centralizer of y.   Z(y) = {x ∈G  |  xy=yx }
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The equivalence class a ̅ of an element a∈G under R is called the #conjugacy class# of a, and consists of all the conjugates of a. That is, a ̅= {  x a x^-1 | x∈G }.
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Our general aim is to count an arbitrary finite group G by counting each conjugacy class separately,
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LEMMA 10.8     Let G be a finite group, and let a∈G. Then the number of  distinct conjugates of a in G is [G:Z(a)]
Let x,y∈G. Then xax^-1 =yay^-1 iff ax^-1 y=x^-1ya  iff  x^-1y∈Z(a)
.
THEOREM 10.9    (Class equation) Let G be a finite group, and let {a1, ... ,ak} consist of one element from each conjugacy class containing at least two elements. Then
|G| = |Z(G)| + [G:Z(a1)] + [G:Z(a1)] + ...+ [G:Z(ak)]
.
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SECTION   11    NORMAL SUBGROUPS
。
DEFINITION
 Let H be a subgroup of G. Then we say that H is a #normal subgroup# if for every h ∈H and g∈G, then ghg^-1∈H. 记为H◁G. 即H◁G={ ghg^-1 |  ∀h ∈H and ∀g∈G}
。
if H◁G, then there is a natural way to turn the set of right ( = left) cosets of H into a group.
H◁G  :  H is a normal subgroup of G
G/H ：quotient group(factor group). read "G mod H." 
If H◁G , then G/H  is  the set of right(=left) cosets of  H in G。 
G/H = {ha | H is a subgroup of G,  ∀h∈H , ∀a∈G}={Ha | ∀a∈G}(这么写正确吗？)
Ha= { ha | H is a subgroup of G,  ∀h∈H , a∈G }
（G/H里每一个元素a都是right(=left) cosets，所以，Ha is a "translate" of H by  element  'a'. There is a natural mapping( homomorphism) ρ from G onto G/H。 ρ(a)=Ha, 然后所有的Ha构成G/H.  ρ is called the #canonical (or natural) homomorphism# from G onto G/H.）
（G/H构成了一个equivalence classes .）
e.g.
let G=(Z, +)and let H=<n>. Then G/H has n  elements, namely H+0, H+1, H+2, ... , H+(n-1).
For instance, if n=6, then (H+2)+(H+5)=H+(2+5)=H+7= H+1, corresponding to 2⊕5= 1 in (Z6 , ⊕)
集合是H，操作符是+，ρ(2)=H+2，
ρ(2)ρ(5) = ρ(2+5)
==> (H+2)+(H+5) = H+(2+5)   ( ρ(a) ($ ρ(b) = ρ(a # b),  ρ(a) = H@a,  这里操作符$),#,@实际上都是+  )

.

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THEOREM 11.1    Let H be a subgroup of G. Then the following are equivalent:
i) H is normal;
ii) gHg^-1 = H for every g∈G;
iii) gH = Hg for every g∈G, i.e., the left cosets are the same as the right cosets.
Condition (iii) does not say that for every h∈H we have gh = hg. Condition (iii) implies that gh will be h' g for some h' ∈H, but it is not necessary for h' to be h.
.
TIIEOREM 11.2    Let G be a group. Then any subgroup of Z( G) is a normal subgroup of G.
COROLLARY          If G is abelian, then every subgroup of G is normal.
THEOREM 11.3    Let H be a subgroup of G such that [ G : H] = 2. Then H is normal in G.
.
Nonabelian groups with the property that every subgroup is normal are called #Hamiltonian#.
every Hamiltonian group must have a subgroup which "looks just like" Q8, 
so Q8 is the simplest Hamiltonian group
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COROLLARY 11.4     Let G be a group, H a subgroup of G, and g∈G. Then gHg^-1  is a subgroup of G, with the same number of elements as H.
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COROLLARY 11.5    If H is a subgroup of G, and no other subgroup has the same number of elements as H, then H is normal in G.
.

Now let H be any subgroup of G. If we want to turn the set of right cosets of H into a group by using an operation that comes from the operation in G, there is really only one way to try to proceed. If we take two right cosets Ha and Hb and try to produce a right coset to be their product, what should it be? Hab, of course.
There is only one problem with this: namely, is it a well-defined operation?(为什么选择这个operation？因为它是G里最直接的operation) That is, suppose Ha is also Ha' and Hb is also Hb'. Is it then true that Hab is the same thing as Ha' b'? In other words, is our intended operation independent of what representatives we pick from the cosets to perform the multiplication? It turns out that it is if and only if H is a normal subgroup  of G.
。
THEOREM 11.6 If  H◁G, then G/H is a group under the operation Ha*Hb=H(ab)
。
|G/H| = [G:H]
|G| = [G:H] * |H| = |G/H| * |H| ，这大概就是之所以被称为‘quotient’的原因
.
quotient groups are very valuable tools. For example, in proving results about finite groups, one often proceeds by induction on the order of the group G. The crucial step is frequently to
consider the quotient of G by some nontrivial normal subgroup. This quotient group is a group of smaller order than G, and hence is subject to the inductive hypothesis.(在某种条件下，研究商群等价与研究G ？)
。
Let H be a subgroup of G. Define N(H)= { g∈G | gHg^- 1 = H}. N(H) is called the #normalizer# of H.
a) Show that N(H) is a subgroup of G. (This is a repeat of Exercise 5.26.)
b) Show that H ◁N(H).
c) Show that if K is a subgroup of G and H ◁ K, then K ⊂= N(H).
.
The #commutator# subgroup G' of G is the smallest subgroup of G containing all the commutators, i.e., all elements of the form x^-1 y^-1 xy . Thus G' is the intersection of all the subgroups that contain all the commutators.
。
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SECTION   12    HOMOMORPHISMS
。
In this section we are going to study "sensible" functions from one group to
another. One reason for doing this is that it reveals something of the dynamic
nature of group theory, something of the way in which different groups can
interact. Another reason is that this interaction often tells us a great deal
about one of the groups involved: One learns about a given group by
bringing into play its relationships with other groups.
The first thing we should do is make clear what the word "sensible"
means in the last paragraph. A function from one group to another is, after
all, a function from one set to another; such a function is "sensible" or
"reasonable" if it takes into account the fact that groups are more than just
sets. Groups have operations on them, and the reasonable functions to look at
are those that behave themselves with respect to these operations.
By this we mean nothing more than the following. Suppose G and H are
groups and φ: G->H is a function. Suppose a,b∈ G. Then we have φ(a)∈ H,
φ(b)∈H, and φ(ab)∈H. What we want is that φ(a)φ(b) should be the same
as φ(ab). That is, we want φ to respect the relationship among the elements
a,b, and ab. We want multiplying a by b and then applying φ to yield the
same element of H as first applying φ to each of a and b and then multiplying
the answers.
Functions that "preserve structure" in this way are called homomorphisms.
。
Let G, H be groups and let φ: G->H be a function. Then φ is called a #homomorphism# if for every a and b in G we have  φ(a)φ(b) = φ(ab)
（或者说，如果G的操作符是#，H的操作符是($，则需满足φ(a) $) φ(b) = φ(a # b)  ）

。
DEFINITIONS Let φ: G->H be a homomorphism. Then φ is called an #isomorphism# if it is a one-to-one onto function. Two groups G and H are said to be isomorphic if there exists an isomorphism from G onto H. If G and H are isomorphic, we write G≅H
。
monomorphisms单一同态:  homomorphism +  one-to-one
epimorphisms外附同态: homomorphism +   onto
isomorphism同形:  homomorphism +  one-to-one + onto
automorphism自同构：an isomorphism from a group onto itself
nontrivial automorphism： automorphism and it is not the identity mapping(It can be shown that any group with more than two elements has a nontrivial automorphism.)
。
TIIEOREM 12.1
i) Let φ: G->H and ψ: H->K be homomorphisms. Then ψoφ : G->K is a homomorphism.
ii) If  φ and ψ are both isomorphisms, so is ψ o φ.
iii) If φ: G->H is an isomorphism, so is φ^-1: H->G.
(the relation of isomorphism is an equivalence relation)
。
TIIEOREM 12.2   Let n be a positive integer, and let G be a cyclic group of order n. Then G≅(Zn, ⊕). Consequently, any two cyclic groups of order n  are isomorphic to each other.
.
THEOREM 12.4    Let φ: G->H be a homomorphism. Then
i) φ(eG)=eH;
ii) for any x∈G and any integer n,  φ(x^n)=[φ(x)]^n;
iii) if o(x)=n, then o[φ(x)] divides n.
.
φ(eG) = φ(a # a^-1) := φ(a) ($ φ(a^-1) =  eH = φ(a) $) φ^-1(a)     ==>    φ(a^-1) = φ^-1(a)
.
THEOREM 12.5   Let φ: G->H be an isomorphism. Then, in addition to the conclusions (i)-(iii) above, we have
iv) o(x)= o[φ(x)], for every x ∈ G;
 v) G and H have the same cardinality;
vi) G is abelian iff  H is.
.
THEOREM 12.7  (Cayley's Theorem)    If G is a group, then G is isomorphic to a subgroup of S($_G$), the symmetric group on the set G
.
A subgroup H of a group G is #characteristic# if φ(H)⊂=H for every  automorphism φ of G.
.
Let G be a group and let g∈G. Show that the mapping φ: G->G given by φ(x)=gxg^-1 is an automorphism of G. It is called an #inner automorphism#, because it comes from an element of G
.
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SECTION   13    HOMOMORPHISMS   AND  NORMAL SUBGROUPS

homomorphisms -----> all normal subgroups (由函数得到元素)
normal subgroups  –---->  all homomorphisms (由元素得到函数)

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DEFINITION
If φ: G->K is a homomorphism, then the #kernel# of φ is  
ker(φ) = φ^-1({eK})= { g∈G | φ(g)=eK}
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TIIEOREM 13.1     For any homomorphism φ: G->K,   ker(φ)◁ G.
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 if we start with H◁ G, form the canonical homomorphism ρ, and then take the kernel, we get back to H. 
if we start with a homomorphism φ: G->K, take its kernel, and then form the canonical  homomorphism ρ: G->G/ker(φ).
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(Fundamental theorem on group homomorphisms) 
Let φ: G->K be a onto-homomorphism from G to K. Then K≅G/ker(φ).
（定义的isomorphism φ‾: G/ker(φ)->K.  
φ‾(Na)=φ(a) ，
φ‾o ρ = φ）







Remark. If φ: G->K is not necessarily onto, we get φ(G)≅G/ker(φ).
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the image of any homomorphism can essentially be recovered from the kernel
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The equation φ‾o ρ = φ tells us that  φ and ρ  (G->K ,  G->G/ker(φ) ) come as close as could be hoped to being the "same" mapping.(因为K≅G/ker(φ))
(群论引导你找到这样的φ ， ρ，φ‾)
。
THEOREM 13.3      Let φ: G->K be an onto homomorphism. There is a one-to-one correspondence between the subgroups of K and those subgroups of G that contain ker(φ), given by H⟼φ(H). We have H◁G iff φ(H)◁K.
Observe that if His any subgroup of G, not necessarily containing ker(φ),
then φ( H) is still a subgroup of K. But it is only by restricting ourselves to the
subgroups of G containing ker(φ) that we get a one-to-one correspondence.
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notation:
If H and K are subgroups of G, then HK={ hk | h∈H, k∈K }
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(Second isomorphism theorem) Let H and K be subgroups of G, and assume K◁G. Then
H/(H∩K)≅ HK/ K
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TIIEOREM 13.5    (Third isomorphism theorem) Suppose H ◁K◁G and H◁G.  Then  K/H◁G/H, and (G/H) / (K/H) ≅ G/K
(◁不可传递)
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SECTION14   DirECT PRODUCTSAND   FINITE ABELIAN  GROUPS

the given group is  isomorphic to the direct product of some of its subgroups. In this way, we can
break the group down into simpler components that are easier to deal with.
Our goal in this section is to use direct products to analyze the structure of finite abelian groups. We will prove a classic result that gives us complete control over these groups.(两个简单子群A，B的直积同构于G，这样可以用更简单的A，B来研究G)

THEOREM 14.1  （如何构造这样的A，B）
Suppose A and B are subgroups of G such that
i) A◁G and B◁G,
ii) AB= G,
iii) A∩B = { e}.
Then G≅AxB.
We first observe that (i)-(iii) imply two more properties of the subgroups A and B:
iv} If ab=a1 b1, where a,a1∈A and b,b1∈B, then a=a1 and b=b1
v) If a∈A and b∈B, then ab= ba.
Observe that, whereas (ii) says that every element of G can be written as ab for some a∈A and b∈ B, (iv) says that this representation is unique.

Examples
1.
Let G= V= {e,a,b,c}, and let A= { e,a} and B= { e,b }. Then A and B are normal, AB= {ee,eb,ae,ab} = {e,b,a,c} =V, and A∩B = { e }. Thus V≅AxB, and since A and B are both isomorphic to Z2 , we have V≅Z2xZ2
。
If p is a prime number and G is a group, then G is said to be of #p-power order# if |G|= p^k for some integer k.
G is called a  #p-group#  if for every x∈G, o(x) is a power of p.
A group is of  #prime-power order#  if it is of p-power order for some prime p.
.
These examples serve to emphasize the fact that two products of nontrivial cyclic groups can be isomorphic without the number of factors being the same, and without the orders of the factors being the same. By restricting ourselves to factors of prime-power order, however, we achieve uniqueness.
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Corollary 14.6    Let G be a finite group with more than two elements. Then  G has a nontrivial automorphism.
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SECTION 15  SYLOW THEOREMS

A group G is called #simple# if its only normal subgroups are { e} and G.
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SECTION    16  RINGS
there are times in real life when one considers both addition and multiplication simultaneously.   We will now consider an abstract notion designed to capture the essence of such situations where two operations interact with each other
What is the essence of the situation for addition and multiplication on Z, for example? If we just look at addition, then we have an abelian group
(Z, + ). If we concentrate on multiplication, we have an associative commuta-
tive binary operation. There happens to be an identity element for ·, but most
elements fail to have inverses. Finally, if we consider both operations at once,
then the most salient point is that they are connected by the distributive laws:
a(b+c)=ab+ac and (b+c)a=ba+ca.
During the nineteenth century, number theorists worked with systems
more inclusive than Z which satisfied the same properties with respect to +
and ·. One motivation for their efforts was the hope that by considering such
systems, one might answer questions about Z that could not be answered by
thinking in terms of  Z alone. Although this hope was only partially realized
(questions about Z can be hard!), a great deal was accomplished, and
moreover, the groundwork was laid for the development of an abstract theory
in the twentieth century
.
Suppose that R is a set and + and · are two binary operations on R. Suppose further that:
i)  (R, +)is an abelian group,
ii) · is associative, and
iii) the distributive laws hold,    i.e.,  a(b+c)=ab+ac and (b+c)a=ba+ca.
Then R, together with the binary operations + and ·, is called a #ring#. We denote it by (R, +, · ), or R for short.
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The #additive identity element# of  R, i.e., the identity element for (R, + ), is
denoted by 0, or OR. If there happens to be an identity element for ·, then it is
an easy matter to see that there is only one such; it is called the #multiplicative 
identity element# or the #unity# of R, and is denoted by l, or lR. A ring that 
possesses a unity is called (what else?) a r#ing with unity#
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DEFINITION    An #integral domain (or just domain, for short)# is a commutative ring with unity in which 1!=0 and there are no nonzero zero-divisors
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THEOREM 16.5  Let R be a ring and let a, b, c ∈R. Assume that a is not a zero-divisor. Then if ab=ac, we have b=c
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Corollary 16.6   Let R be a commutative ring with unity 1!=0. Then R is an integral domain iff whenever a, b, c ∈R satisfy ab= ac and a!=0,. we have b=c.
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DEFINmONS   R is called a #division ring# if R has a unity 1 !=0 and every nonzero element of R is a unit. A commutative division ring is called a #field#.

Another way of saying that R is a division ring is to say that the set
R- {0} forms a group under multiplication. Saying that R is a field amounts
to saying that this group is abelian.
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THEOREM 16.7   Every finite integral domain is a field.
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Of course, there exist infinite domains that are not fields---- Z for instance. On the other hand, every field, finite or infinite, is a domain, because units are not zero-divisors. Thus the notions of "domain" and "field" coincide for finite rings, but, in general, "field" is stronger.


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SECTION  17 SUBRINGS, IDEALS, AND QUOTIENT RINGS

Let (R, +, ·) be a ring. A subset S of R is called a #subring# of R if the elements of S form a ring under + and ·
In particular, the definition requires that (S, +) be a subgroup of (R, + ).
Thus ifS is a subring of R, then we know that the additive identity element 0
of R is in S, and that S is closed under addition and under additive inverses.
The relationship between R and S with respect to multiplication need not be
so clean.（为什么会这样？因为+和·的‘地位/角色’不同？这个‘地位/角色’是由(Ring定义里的)分配律决定的） For example, if R has a multiplicative identity 1, then 1 need not be
in S, and it is even possible that S may have an identity element different
from that of R.
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TEOREM 17.1      Let (R, +,·)be a ring, and letS be a subset of R. Then S is a subring of R iff the following two conditions are satisfied:
i) (S, +) is a subgroup of (R, + ); and
ii) S is closed under multiplication, that is, if  r1,r2 ∈S then r1r2∈S
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COROLLARY    17.2     Let (R, +, ·) be a ring and let S be a nonempty subset of R.
Then S is 3: subring of R iff the following two conditions hold:
i) for every r1 ,r2∈S, we have r1-r2∈S;
ii) for every r1 ,r2∈S, we have r1 r2∈S.
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DEFINitON    A subring S of a ring R is called an #ideal# of R if for every s∈S and r∈R we have rs∈S and sr∈S.
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R / S is called the #quotient ring (or factor ring)# of R by S.
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12. Let R be a commutative ring with unity, and let a∈R. Let
aR={ar | r∈R}
so that aR is the set of all multiples of a in R. Then aR is an ideal of R.
We call aR the #principal ideal generated by a#
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SECTION18
RING HOMOMORPHISMS

DEFINmON    Let R and S be rings, and let φ: R->S be a function. Then φ is called a #(ring) homomorphism# if for every a, b∈R we have
i) φ( a + b) = φ( a) + φ( b) and
ii) φ( ab)= φ( a )φ( b).
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COROLLARY 19.4 Let F be a field, f(X) a polynomial of degree n over F. Then
f(X) has at most n distinct roots in F.
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However, our point of
view at the moment is that f(X) is a formal expression, and not the function
this expression induces on R. This distinction is a significant one, because it is
possible for two different polynomials to induce the same function on R. For
example, if R is (Z2 , ⊕, ⊙), then both 0+0X +0X 2 + · · · and 0+ 1X+ 1X^2
give us the function that maps every element of R to 0.
.
We thus define the #content# of a nonzero polynomial in Z[X] to be the
g.c.d. of its coefficients; we call a polynomial #priniitive# if its content is 1.
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how polynomial rings can be used to construct interesting new fields

P201 Example 1,  Example 2 如何解方程
。
THEOREM 20.3   (Kronecker) Let F be a field and let f(X)∈F[X] have degree >=1. Then there is an extension field K of F that contains a root of f(X).


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