<Contemporary Abstract Algebra> (by Joseph Gallian) 笔记

<Contemporary Abstract Algebra> by Joseph Gallian

Definition Equivalence Relation
An equivalence relation on a set S is a set R of ordered pairs of
elements of S such that
1. (a, a) ∈R for all a∈S (reflexive property).
2. (a, b) ∈R implies (b, a) ∈R (symmetric property).
3. (a, b) ∈R and (b, c)∈R imply (a, c)∈R (transitive property).
.
If ~ is an equivalence relation on a set S and a∈S, then the set [a]= {x∈S | x ~a} is called the
#equivalence class# of S containing a.
.
Definition Partition
A partition of a set S is a collection of nonempty disjoint subsets of S whose union is S.
.
Theorem 0.7 Equivalence Classes Partition(equivalence relations 和 partitions之间的关系)
The equivalence classes of an equivalence relation on a set S
constitute a partition of S. Conversely, for any partition P of S, there
is an equivalence relation on S whose equivalence classes are the
elements of P.
(简单地理解，假设partition A和B有公共元素c，则由于equivalence 的传递性(传染性)，则A=B. 这与A，B是不同的partition矛盾。)
。
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1   Introduction to Groups
The corresponding group is denoted by Dn and is called the #dihedral group of order 2n#。 （or group of symmetries of a regular n-gon）
正方形对应D4
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2 group
A binary operation on a set G, then, is simply a method (or formula) by which the members of an ordered pair from G combine to yield a new member of G. This condition is called #closure#.
The best way to grasp the meat of a theorem is to see what it says in specific cases.
U(n)：all positive integers less than n and relatively prime to n. 
φ(n):=|U(n)|.   φ(n) is called the Euler phi function
。
The goal of abstract algebra is to discover truths about #algebraic
systems# (that is, sets with one or more binary operations) that are inde-
pendent of the specific nature of the operations. All one knows
or needs to know is that these operations, whatever they may be, have
certain properties. We then seek to deduce consequences of these
properties. This is why this branch of mathematics is called #abstract#
algebra. It must be remembered, however, that when a specific group
is being discussed, a specific operation must be given (at least
im plicitly).

We are forced to restrict ourselves to the properties that all groups have; that is, we must view groups as abstract entities rather than argue by example.
。
Theorem 2.4 Socks–Shoes Property
For group elements a and b, (ab) ^-1 = b ^-1 a ^-1 .
.
H ≤G :  H is a subgroup of G
H <G:H is a proper subgroup of G
.
Theorem 3.1 One-Step Subgroup Test
Let G be a group and H a nonempty subset of G. If ab 21 is in H
whenever a and b are in H, then H is a subgroup of G. (In additive
notation, if a 2 b is in H whenever a and b are in H, then H is a
subgroup of G.)
.
Theorem 3.2 Two-Step Subgroup Test
Let G be a group and let H be a nonempty subset of G. If 
- ab is in H whenever a and b are in H (H is closed under the operation), and 
- a^-1 is in H whenever a is in H (H is closed under taking inverses), 
then H is a subgroup of G.
.
(如何建立一个subgroup)Let G be an Abelian group and H and K be subgroups of G. Then HK={hk | h∈H, k∈K} is a subgroup of G.
.
How do you prove that a subset of a group is not a subgroup? Here
are three possible ways, any one of which guarantees that the subset is
not a subgroup:
1. Show that the identity is not in the set.
2. Exhibit an element of the set whose inverse is not in the set.
3. Exhibit two elements of the set whose product is not in the set.
.
Theorem 3.3 Finite Subgroup Test
Let H be a nonempty finite subset of a group G. If H is closed under the operation of G, then H is a subgroup of G.
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Theorem 3.4 <a> Is a Subgroup
For any element a of a group G, it is useful to think of kal as the smallest subgroup of G containing a.
.
Definition Center of a Group（群的Center）
The center, Z(G), of a group G is the subset of elements in G that commute with every element of G. In symbols,
Z(G) ={a∈G | ax=xa for all x in G}.
.
Z(G) is a subgroup of G.
。
Definition Centralizer of a in G（元素的Centralizer。）
Let a be a fixed element of a group G. The centralizer of a in G, C(a), is the set of all elements in G that commute with a. In symbols, C(a) = {g∈G | ga = ag, a∈G}
.
Z(G) = ∪C(ai) (不同元素的Centralizer的并集构成Center)
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Theorem 3.6 C(a) Is a Subgroup
--------------------------------------------
4 Cyclic Groups
Theorem 4.1 Criterion for a^i =a^j
Let G be a group, and let a belong to G. 
If a has infinite order, then a^i=a^j if and only if i=j. 
If a has finite order, say, n, then <a>={e, a, a^2 , . . . , a^n–1} and a^i=a^j if and only if n divides i – j.
.
Corollary 1 |a| = |<a>|
Corollary 2 a^k=e Implies That |a| Divides k
.
For these reasons, the cyclic groups Zn and Z serve as prototypes for all cyclic groups, and algebraists say that there is essentially only one cyclic group of each order. What is meant by this is that, although
there may be many different sets of the form {a^n | n∈Z}, there is essentially only one way to operate on these sets. 
Algebraists do not really care what the elements of a set are; they care only about the algebraic properties of the set—that is, the ways in which the elements of a set can be combined.
.
Theorem 4.2   <a^k> = <a^gcd(n,k) > and |a^k | = n/gcd(n, k)
The advantage of Theorem 4.2 is that it allows us to replace one generator of a cyclic subgroup with a more convenient one.
.
cyclic groups play the role of building blocks for all finite Abelian groups in much the same way that primes are the building blocks for the integers
.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Given two permutations a and b, express ab in a disjoint cycle form.
a= (12)(3)(45), b =(153)(24)
==> ab = (12)(3)(45)(153)(24)
[iterator 0] begin with element 1
(2,4) fixes 1, (153) sends 1 to 5, (45) sends 5 to 4, (3)fixes 4, (12) fixes 4, 
so, [iterator 0] sends 1 to 4
==> ab=(14...)...
[iterator 1] begin with element 4
(2,4) sends 4 to 2, (153) fixes 2, (45) fixes 2, (3)fixes 2, (12) sends 2 to 1, 
so, [iterator 1] sends 4 to1
==> ab=(141...)...==>ab=(14)...
[iterator 2] since 1 and 4 are selected, they are ignored. We selected the minimum for 2,3,5,6, it is 2.
[iterator 2] begin with element 2
(2,4) sends 2 to 4, (153) fixes 4, (45) sends 4 to 5, (3)fixes 5, (12) fixes 5, 
so, [iterator 1] sends 2 to5
==>ab=(14)(25...)...
[iterator 3] begin with element 5
(2,4) fixes 5, (153) sends 5 to 3, (45) fixes 3, (3)fixes 3, (12) fixes 3, 
so, [iterator 3] sends 5 to3
==>ab=(14)(253...)...
[iterator 4] begin with element 3
(2,4) fixes 3, (153) sends 3 to 1, (45) fixes 1, (3)fixes 1, (12) fixes 2,
so, [iterator 3] sends 3 to2
==>ab=(14)(2532...)... ==>ab=(14)(253)...
[iterator 4] since 1,2,3,4,5 are selected, they are ignored. We selected the minimum for 6, it is 6.
[iterator 4] begin with element 6
(2,4) fixes 6, (153) fixes 6, (45) fixes 6, (3)fixes 6, (12) fixes 6,
==>ab=(14)(253)(6)
and (6) can be ingored
==>ab=(14)(253)
(DONE)
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Theorem 5.3 Order of a Permutation (Ruffini, 1799 )
The order of a permutation of a finite set written in disjoint cycle form is the least common multiple of the lengths of the cycles.(P107 Example5)
.
Theorem 5.4 Product of 2 -Cycles
Every permutation in Sn , n>1, is a product of 2-cycles.
.
Theorem 5.6 Even Permutations Form a Group
The set of even permutations in Sn forms a subgroup of Sn
.
The subgroup of even permutations in S n arises so often that we give
it a special name and notation.
Definition Alternating Group of Degree n
The group of even permutations of n symbols is denoted by A n and is
called the alternating group of degree n
The names for the symmetric group and the alternating group of degree
n come from the study of polynomials over n variables. A symmetric
polynomial in the variables x1 , x2 , . . . , xn is one that is unchanged under
any transposition of two of the variables. An alternating polynomial is
one that changes signs under any transposition of two of the variables.
.
几个不错的例子：
P113 EXAMPLE 9 (Loren Larson) A Sliding Disk Puzzle
P114 Rubik's Cube
P115 A Check-Digit Scheme Based on D5
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6 Isomorphisms
In this chapter, we give a formal method for determining whether two groups defined in different terms are really the same. When this is the case, we say that there is an isomorphism between the two groups.
group里元素是什么不重要。重要的是元素与元素之间的关系。如果两个群A里的元素满足一个关系，群B里的元素也满足同样的关系。就认为A与B isomorphic。A≈B
。
Check  isomorphic:
Step 1 “Mapping.” Define a candidate for the isomorphism; that is, define a function φ from G to G.
Step 2 “1–1.” Prove that f is one-to-one; that is, assume that φ(a)=φ(b) and prove that a=b.
Step 3 “Onto.” Prove that f is onto; that is, for any element g in G, find an element g in G such that φ(g)=g.
Step 4 “O.P.” Prove that f is operation-preserving; that is, show that φ(ab)=φ(a)φ(b) for all a and b in G.
.(Iso(A,B))
Theorem 6.1 Cayley’s Theorem (1854)
Every group is isomorphic to a group of permutations.
.
Why is  Cayley’s Theorem so important?
- it allows us to represent an abstract group in a concrete way.
- it shows that the present-day set of axioms we have adopted for a group is the correct abstraction of its much earlier predecessor—a group of permutations.
-It tells us that abstract groups are not different from permutation groups. 
Rather, it is the viewpoint that is different. It is this difference of viewpoint that has stimulated the tremendous progress in group theory and many other branches of mathematics in the 20th century.
.
G‾={gx | g∈G,  ∀x∈G}  is called the left regular representation of G
.
Theorem 6.2 Properties of Isomorphisms Acting on Elements
Suppose that f is an isomorphism from a group G onto a group G‾. Then
1. f carries the identity of G to the identity of G‾.
2. For every integer n and for every group element a in G, f(a^n )=[f(a)]^n .
3. For any elements a and b in G, a and b commute iff f(a) and f(b) commute.
4. G=<a> iff G‾=<f(a)>
5. |a| =|f(a)| for all a in G (isomorphisms preserve orders).
6. For a fixed integer k and a fixed group element b in G, the equation x^k=b has the same number of solutions in G as does the equation x^k = f(b) in G‾.
7. If G is finite, then G and G‾ have exactly the same number of elements of every order.
Theorem 6.3 Properties of Isomorphisms Acting on Groups
Suppose that f is an isomorphism from a group G onto a group G. Then
- f^-1 is an isomorphism from G‾ onto G.
- G is Abelian iff G‾ is Abelian.
- G is cyclic iff G‾ is cyclic.
- If K is a subgroup of G, then f(K)={f(k) | k∈K} is a subgroup of G‾.
- If K‾ is a subgroup of G‾, then f^-1(K) ={g∈G | f(g) ∈K‾} is a subgroup of G.
- f(Z(G)) =Z(G‾).
.
Definition Inner Automorphism Induced by a
Let G be a group, and let a∈G. The function f($_a$) defined by f($_a$) (x) = axa^-1 for all x in G is called the #inner# automorphism of G induced by a
.
Definition Automorphism
An isomorphism from a group G onto itself is called an automorphism of G.  (  Aut(G) := Iso(G,G)  )
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Theorem 6.4 Aut(G) and Inn(G) Are Groups †
Theorem 6.5 Aut(Zn ) ≈ U(n)
----------------------------------------------------
7 Cosets and Lagrange’s Theorem
Definition Coset of H in G
Let G be a group and let H be a nonempty subset of G. For any a∈G,
the set {ah | h∈H} is denoted by aH. Analogously, Ha ={ha | h [ H}
and aHa^-1={aha^-1 | h∈H}. When H is a subgroup of G, the set aH is
called the left coset of H in G containing a, whereas Ha is called the right
coset of H in G containing a. 
In this case, the element a is called the #coset representative# of aH (or Ha). We use |aH| to denote the number of elements in the set aH, and |Ha| to denote the number of elements in Ha
.
First, cosets are usually not subgroups.
Second, aH may be the same as bH, even though a is not the same as b.
Third, aH need not be the same as Ha
.
Lemma Properties of Cosets
Let H be a subgroup of G, and let a,b∈G. Then,
1.  a∈aH.
2.9  aH=H     iff    a∈H     iff    aH is a subgroup of G.
3.  (ab)H=a(bH) and H(ab)=(Ha)b.
4.6  aH=bH    iff   a∈bH   iff    a^-1 b∈H
5.  aH=bH or aH∩bH=⊘.
7.  |aH| = |bH|.
8.  aH= Ha   iff   H=aHa^-1 .
Property 4 shows that a left coset of H is uniquely determined by any one of its elements. In particular, any element of a left coset can be used to represent the coset.
Property 5 says that two left cosets of H are either identical or disjoint. Thus, a left coset of H is uniquely determined by any one of its elements.
Property 6 shows how we may transfer a question about equality of left cosets of H to a question about H itself and vice versa. 
Property 7 says that all left cosets of H have the same size.
Property 8 is analogous to property 6 in that it shows how a question about the equality of the left and right cosets of H containing a is equivalent to a question about the equality of two subgroups of G. 
Property 9 says that H itself is the only coset of H that is a subgroup of G.
.
we may view the cosets of H as a partitioning of G into equivalence classes under the equivalence relation defined by a~b if aH=bH
.
For example, if G is 3-space R 3 and H is a plane through the origin, then the coset (a, b, c) +H (addition is done componentwise) is the plane passing through the point (a, b, c) and parallel to H. Thus, the cosets of H constitute a partition of 3-space into planes parallel to H.
.
Theorem 7.1 Lagrange’s Theorem : |H| Divides |G|
If G is a finite group and H is a subgroup of G, then |H| divides |G|.
Moreover, the number of distinct left (right) cosets of H in G is |G|/|H|.
.
e.g. a group of order 12 may have subgroups of order 12, 6, 4, 3, 2, 1, but no others.
.
Corollary 1 |G:H|=|G|/|H|. The index of a subgroup H in G
Corollary 2 |a| Divides |G|.  |a| means the order of element a
Corollary 3 Groups of Prime Order Are Cyclic
Corollary 4 a^ |G| = e
Theorem 7.2 |HK| = |H||K|/|H∩K|
.
An Application of Cosets to Permutation Groups
Definition Stabilizer of a Point
Let G be a group of permutations of a set S. For each i in S, let stab G (i) ={f∈G | f(i)=i}. We call stab($_ G$)(i) the stabilizer of i in G.
.
stab($_ G$)(i) is a subgroup of G
.
Definition Orbit of a Point
Let G be a group of permutations of a set S. For each s in S, let orb G (s)={f(s) | f ∈G}. The set orb($_G$)(s) is a subset of S called the orbit of s under G. We use |orb($_G$) (s)| to denote the number of elements in orb G (s).
.
EXAMPLE 7 Let G = {(1), (132)(465)(78), (132)(465), (123)(456), (123)(456)(78), (78)}.
Then,
orb ($_G$) (1) = {1, 3, 2},
orb ($_G$) (2) = {2, 1, 3},
orb ($_G$) (4) = {4, 6, 5},
orb ($_G$) (7) = {7, 8},
stab ($_G$) (1) = {(1), (78)},
stab ($_G$) (2) = {(1), (78)},
stab ($_G$) (4) = {(1), (78)},
stab ($_G$) (7) = {(1), (132)(465), (123)(456)}.
.
Theorem 7.4 Orbit-Stabilizer Theorem
Let G be a finite group of permutations of a set S. Then, for any i from S, |G| = |orb($_G$) (i)| * |stab ($_G$) (i)|.
.
Let A ={1, 2, . . . , n}. The set of all permutations of A is called the symmetric group of degree n and is denoted by Sn .
.
Theorem 7.5 The Rotation Group of a Cube
The group of rotations of a cube is isomorphic to S4
.
---------------------------------------------------
8  External Direct Products
Definition External Direct Product
G1⊕G2⊕...⊕Gn = {(g1, g2 , . . . , gn ) | gi∈Gi },
.
Corollary 2 Criterion for Z$_{n1n2 . . . nk}$ ≈ Zn1⊕ Zn2⊕ . . .⊕ Znk
Let m =n1n2 . . . nk . Then Zm is isomorphic to Zn1⊕ Zn2⊕ . . .⊕ Znk if and only if ni and nj are relatively prime when i != j
.
If G=H⊕K, then |G|= |H||K|;
every element of G has the form (h, k) where h∈H and k∈K; 
if |h| and |k| are finite, then |(h, k)| =lcm(|h|, |k|);
if H and K are Abelian, then G is Abelian; 
if H and K are cyclic and |H| and |K| are relatively prime, then H⊕K is cyclic
.
Applications
             - Data Security
             - Public Key Cryptography
             - Digital Signatures
             - Genetics
             - Electric Circuits
----------------------------------------------------
9  Normal Subgroups  and   Factor Groups
。
Definition Normal Subgroup
H∈sub(G) is called a #normal subgroup# of G if aH=Ha for all a in G. We denote this by H◁G.
(H=(  K∈sub(G),    if ∀k∈K, ∀a∈G, aka^-1∈K then return K, else return ⊘  )这么写对吗？)
You should think of a normal subgroup in this way: You can switch
the order of a product of an element a from the group and an element h
from the normal subgroup H, but you must “fudge” a bit on the element
from the normal subgroup H by using some h' from H rather than h.
That is, there is an element h' in H such that ah = h'a. Likewise, there
is some h'' in H such that ha=ah''. (It is possible that h' = h or h'' = h,
but we may not assume this.)
.
Theorem 9.1 Normal Subgroup Test
A subgroup H of G is normal in G iff xHx^-1 ⊆H for all x in G.
.
Every subgroup of an Abelian group is normal.
.
(用 normal subgroup 来构造新的subgroup)：Let H be a normal subgroup of a group G and K be any
subgroup of G. Then HK ={hk | h∈H, k∈K} is a subgroup of G.
。
When  H◁G, then the set of left (or right) cosets of H in G is itself a group—called the factor group(or quotient group) of G by H， 记为G/H . 
one can obtain information about a group by studying one of its factor groups.(quotient group之所以重要的原因)
。
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
2016.6.20
<<Persistent Homology: An Introduction and a New Text Representation for Natural Language Processing>>里的一些介绍很不错
Definition 4.
Given a subgroup H of an abelian group G, for any a∈G, the set a∗H = {a∗h | h∈H} is the coset of H represented by a.
Definition 7. 
The kernel of a homomorphism φ:G→G' is kerφ = {a∈G | φ(a)=e'}. 
In other words, the kernel is the elements that map to identity.
Define a new binary operation ☆ not on the elements of G but on the cosets of H:
(a∗H) ☆ (b∗H) = (a∗b)∗H, ∀a,b∈G.
Definition 8. 
The cosets {a∗H |a∈G} under the operation ☆ form a group, called the quotient group G/H.
quotient group G/H： <{a∗H |a∈G},☆>
- coset是一个set，(set里的元素是G里的元素)
- quotient group G/H里的每一个元素是(H的)一个coset
Persistent homology finds “holes” by identifying equivalent cycles: Consider the following space in yellow with a small white hole. Imagine the blue cycle as a rubber band. It can be stretched and bent within the space into the green cycle, but not the red one without tearing itself. 
There are two equivalent classes of rubber bands: some surround the hole and others do not. Conversely, two equivalent classes indicate one hole. 

Because kerφ is a subgroup, we can partition G into cosets of the form a∗kerφ for a∈G. These cosets are the white or blue squares.
It is useful to think of quotient groups as “higher level” groups defined on the squares in the previous picture. kerφ (the blue square) is a subgroup of G. The elements of the quotient group G/kerφ are the cosets of kerφ, i.e. all the squares.
Group theory is important because when counting “holes” in homology, G will be the group of cycles (the rubber bands).
The blue square will be the subgroup of “uninteresting rubber bands” that do not surround holes, similar to the earlier blue and green rubber bands. The quotient group “all rubber bands”/“uninteresting rubber bands” will identify holes.
However, the rubber bands are continuous and difficult to compute. We first need to discretize the space into a simpler structure called simplicial complex.
The intuition of simplicial complex is that if a simplex is in K, all its faces need to be in K, too. In addition, the simplices have to be glued together along
whole faces or be separate.
Simplicial complex plays the role of the yellow space in the
rubber band example. We next introduce the discrete version
of the rubber bands.
Definition 14. 
A p-chain is a subset of p-simplices in a simplicial complex K.
Definition 17. 
The boundary of a p-chain is the +2 sum of the boundaries of its simplices. Taking the boundary is a group homomorphism ∂p from Cp to Cp−1.
Note faces shared  by an even number of p-simplices in the chain will cancel  out:

the boundary of any higher order (p+1)-chain is always a p-cycle. For example, the left figure below shows a simplicial complex containing a (p+1) = 2 chain (the yellow triangle). Its boundary c1 (blue) is indeed a 1-cycle

Definition 19. 
A p-boundary-cycle is a p-cycle that is also the boundary of some (p+1)-chain.
Let Bp = ∂($_{p+1}$)C($_{p+1}$), namely all the p-boundary-cycles.
Bp are the uninteresting rubber bands. In the example above, B1 = {0, c1}, none surrounding any holes.
c2,c3 are interesting because they surround the hole in the rectangle. In fact, we can drag the rubber band c2 over the yellow triangle and turn it into
c3. Formally, we do this by c3 = c2 + c1 . 
Intuitively, c2 and c3 are equivalent in the hole they surround. 
More generally, such equivalence class is obtained by c + Bp: we are allowed to drag a p-cycle rubber band c over any (p + 1)-simplices without changing the holes (or the lack thereof) it surrounds.
Returning to the example, we now see all the 1-cycles for this simplicial complex: Z1 = {0, c1, c2, c3}. The uninteresting ones are B1 = {0, c1}, a subgroup of Z1 . The interesting ones are c2+B1= c3+B1 = {c2, c3}: this should remind us of cosets and quotient group.(B1是子群H，操作符是+，c2+B1是coset，c3+B1也是coset；{c2, c3}是商群)
Definition 20. 
The p-th homology group is the quotient group Hp=Zp/Bp. The p-th Betti number is its rank: βp=rank(Hp).
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Theorem 9.2 Factor Groups (O. Hölder, 1889)
Let G be a group and H◁G. The set G/H = {aH | a∈G} is a group under the operation (aH)(bH) = abH.
。
if the correspondence aHbH=abH defines a group operation on the set of left cosets of H in G, then H◁G.
。
|aH|：
- the size of the set aH
- think aH as an element of G/H, and |aH| as the order of the element aH in G/H.
.
one can see that the formation of a factor group G/H causes a systematic collapse of the elements of G. In particular, all the elements in the coset of H containing ‘a’ collapse to the single group element aH in G/H.
.
Applications of Factor Groups
Why are factor groups important? Well, when G is finite and H != {e},
G/H is smaller than G, and its structure is usually less complicated than
that of G. At the same time, G/H simulates G in many ways. In fact, we
may think of a factor group of G as a less complicated approximation
of G (similar to using the rational number 3.14 for the irrational
number p). What makes factor groups important is that one can often
deduce properties of G by examining the less complicated group G/H
instead.
.
Theorem 9.3 G/Z Theorem
Let G be a group and let Z(G) be the center of G. If G/Z(G) is cyclic, then G is Abelian
Theorem 9.4 G/Z(G) ≈Inn(G)
Theorem 9.5 Cauchy’s Theorem for Abelian Groups
Let G be a finite Abelian group and let p be a prime that divides the order of G. Then G has an element of order p.
.
Internal Direct Products
Definition Internal Direct Product of H and K
We say that G is the internal direct product of H and K and write G = H⨯K
 if H and K are normal subgroups of G and G = HK and H ∩ K = {e}.
.
 if H∈sub(G),K∈sub(G), G≈H⊕K; then G=HxK,
.
Internal direct product can be formed within G itself, using subgroups of G and the operation of G. External direct product can be formed with totally unrelated groups by creating a new set and a new
operation.
.
Definition Internal Direct Product  H1⨯H2⨯... ⨯Hn
Let H1◁G , H2◁G , . . . , Hn◁G. We say that G is the internal direct product of H1,H2, . . . , Hn and write G = H1⨯H2⨯... ⨯Hn , if
1. G = H1H2 ... Hn ={h1h2...hn | hi∈Hi },
2. (H1H2...Hi ) > Hi+1={e} for i = 1,2, . . . , n-1.
Motivation of this definition:
We want H1⨯H2⨯... ⨯Hn ≈ H1⊕H2⊕... ⊕Hn.
.
Theorem 9.7 Classification of Groups of Order p^2
If |G|= p^2 , where p is a prime,  then G≈ Zp^2 or G ≈ Zp⊕Zp .
Corollary
If |G|= p^2 , where p is a prime, then G is Abelian.
.
--------------------------------------------------
10  Group  Homomorphisms
Definition Group Homomorphism
A homomorphism f from a group G to a group G‾ is a mapping from G into G‾ that preserves the group operation; that is, f(ab) =f(a)f(b) for all a, b in G 
.
Definition Kernel of a Homomorphism
The kernel of a homomorphism f:G --> G‾ is the set {x∈G | f(x) = e($_{G‾}$)}. The kernel of f is denoted by Ker(f).
.
every linear transformation is a group homomorphism and the null-space is the same as the kernel. 
An invertible linear transformation is a group isomorphism.
.
Theorem 10.1 Properties of Elements Under Homomorphisms
Let f be a homomorphism from a group G to a group G‾ and let g∈G. Then
1. f carries the identity of G to the identity of G‾.
2. f(g^n ) =(f(g)) ^n for all n in Z.
3. If |g| is finite, then |f(g)| divides |g|.
4. Ker(f) ∈sub(G).
5. f(a) = f(b) iff aKer(f)= bKer(f).
6. If f(g)=g', then f^-1(g')={x∈G | f(x)=g'}=gKer(f). (应用：微分方程组有特解x，对应的齐次方程组的通解S，则x+S是非齐次方程组的通解。)
.
Theorem 10.2 Properties of Subgroups Under Homomorphisms
Let f be a homomorphism from a group G to a group G‾ and let H∈sub(G). Then
1. f(H) = {f(h) | h∈H} is a subgroup of G‾.
2. If H is cyclic, then f(H) is cyclic.
3. If H is Abelian, then f(H) is Abelian.
4. If H◁G, then f(H)◁f(G).
5. If |Ker f|=n, then f is an n-to-1 mapping from G onto f(G).
6. If |H|=n, then |f(H)| divides n.
7. If K‾∈sub(G‾), then f ^-1 (K‾) ={k∈G | f(k) ∈K‾} is a subgroup of G.
8. If K‾◁G, then f^-1(K‾)={k∈G | f(k)∈K‾}◁G.
9. If f is onto and Ker(f) ={e}, then f is an isomorphism from G to G‾.
.
Corollary Kernels Are Normal Subgroups
If f: G->G‾, then Ker(f)◁G
.

Theorem 10.3 First Isomorphism Theorem (Jordan, 1870)
Let f be a group homomorphism from G to G‾. Then the mapping ψ:G/Ker(f) --> f(G)  given by ψ( gKer(f) )=f(g), is an isomorphism.
In symbols, G/Ker(f) ≈ f(G).
.

f is a  homomorphism.
ψ is an isomorphism. G/Ker(f) ≈ f(G).
r: G-->G/Ker f is defined as r(g) = gKer(f)
The mapping r is  called the natural mapping from G to G/Ker(f)
.
N(H)/C(H)≈Aut(H)
H∈sub(G)
N(H)={x∈G | xHx^-1=H}
C(H)={x∈G | xhx^-1=h, ∀h∈H}
.
Theorem 10.4 Normal Subgroups Are Kernels
N◁G iff N is Ker(f), f: G->G/N, f(g) = gN
.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
对 First Isomorphism Theorem的通俗解释：
But what about homomorphisms in general? Why would one care to study a homomorphism of a group? 
The answer is that, just as was the case with G/Ker(f), f(G) tell us some of the properties of G . One measure of the likeness(我觉得更像是unlikeness) of G and f(G) is the size of Ker(f). 
- If the Ker(f) is the identity, then f(G) tells us everything (group theoretically) about G (the two being isomorphic(因为f由homomorphism变成了isomorphism) ). 
- If Ker(f) is G itself, then f(G) tells us nothing about G.
(个人理解：G经过'相似'变换f变成为f(G), G里的很多元素gi被‘相似’成为f(gi), 但G里仍有些元素ki无法被‘相似’变换过去，只能得到f(ki)=e($_{f(G)}$)(这本质上是由group的封闭性导致的).  所以有理由认为ki没有被‘相似’变换到f(G)。  
所以，如果ki的数量越多，我们就有理由认为G和f(G)越不相似。ki组成的集合就是Ker(f).   
相反，如果{ki}={e}时，就认为G里的所有元素gi都被相似变换过去了，所以说G和f(G)相似度很高，此时这个相似变换f是isomorphic )
Between these two extremes, some information about G is preserved and some is lost.
The utility of f lies in its ability to preserve the group properties we want, while losing some inessential ones. In this way, we have replaced G by  f(G)  less complicated (and therefore easier to study) than G; but, in the process, we have saved enough information to answer questions that we have about G itself.
Although an isomorphism is a special case of a homomorphism, the
two concepts have entirely different roles. Whereas isomorphisms
allow us to look at a group in an alternative way, homomorphisms act as
investigative tools. 
The following analogy between homomorphisms
and photography may be instructive. A photograph of a person cannot
tell us the person’s exact height, weight, or age. Nevertheless, we may
be able to decide from a photograph whether the person is tall or short,
heavy or thin, old or young, male or female. In the same way, a homo-
morphic image of a group gives us some information about the group.
In certain branches of group theory, and especially in physics and
chemistry, one often wants to know all homomorphic images of a group
that are matrix groups over the complex numbers (these are called group
representations). Here, we may carry our analogy with photography one
step further by saying that this is like wanting photographs of a person
from many different angles (front view, profile, head-to-toe view, close-
up, etc.), as well as x-rays! Just as this composite information from the
photographs reveals much about the person, several homomorphic im-
ages of a group reveal much about the group.
。
2016.4.21
P285  
对 isomorphism and homomorphism的通俗解释：
Again as with group theory, the roles of isomorphisms and homomorphisms are entirely distinct.  
An isomorphism is used to show that two rings are #algebraically identical#; 
a homomorphism is used to #simplify# a ring while #retaining certain of its features#.
上面说的“如果{ki}={e}时”，可认为在 ”simplify“的时候ki的信息丢失了
。
2016.4.27
First Isomorphism Theorem已经很NB地刷新了我的世界观，但这还不是结束：
First Isomorphism Theorem研究对象是group之间的functions。如果不考虑group的操作符，那么group就变成了set(这扩大了研究范围)，然后First Isomorphism Theorem就被推广成为category theory里的一个定理(<Algebra:Chapter 0>, P15, Theorem 2.7). 
所以有理由把category theory看作是群论的推广----群论研究的是group之间的functions，category theory研究的是set之间的functions。
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
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11  Fundamental   Theorem of Finite  Abelian Groups
Theorem 11.1 Fundamental Theorem of Finite Abelian Groups
Every finite Abelian group is a direct product of cyclic groups of prime-power order. Moreover, the number of terms in the product and the orders of the cyclic groups are uniquely determined by the group.
。
determining the isomorphism class of G
。
the uniqueness portion of the Fundamental Theorem
guarantees that distinct partitions of k yield distinct isomorphism
classes. Thus, for example, Z9⊕Z3 is not isomorphic to Z3⊕Z3⊕Z3 .
.
A reliable mnemonic for comparing external direct products is the can-
cellation property: If A is #finite#, then  A⊕B≈A⊕C  iff  B≈C
。
Corollary Existence of Subgroups of Abelian Groups
If m divides the order of a finite Abelian group G, then G has a
subgroup of order m.
。
------------------------------------------------
PART 3  Rings
------------------------------------------------
12  Introduction to Rings
Definition Ring
A #ring# R is a set with two binary operations, addition (denoted by a + b) and multiplication (denoted by ab), such that for all a, b, c in R:
1. a + b = b + a.（+的交换率）
2. (a + b) + c = a + (b + c).（+的结合率）
3. There is an additive identity 0. That is, there is an element 0 in R such that a + 0 = a for all a in R.（+的零元）
4. There is an element -a in R such that a + (-a) = 0.（+的逆元）
5. a(bc) = (ab)c.（*的结合率）
6. a(b + c) = ab + ac and (b + c) a = ba + ca.(左分配，右分配)
(R, +, · )
- 没有定义 multiplication的单位元。<----> unity
- multiplication need not be commutative <---->   commutative ring
- 'a' is a #unit# if a^-1 exists
.
A #unity# (or identity) in a ring is a nonzero element that is an identity under multiplication. 
'a' is a #unit# if a^-1 exists.
.
。
Theorem 12.1 Rules of Multiplication
Let a, b, and c belong to a ring R. Then
1.a0 = 0a = 0.
2.a(-b) =(-a)b = -(ab).
3.(-a)(-b) = ab. 
4.a(b- c) = ab - ac and (b- c)a = ba - ca.
Furthermore, if R has a unity element 1, then
5. (-1)a =-a.
6. (-1)(-1) = 1.
。
Theorem 12.2 Uniqueness of the Unity and Inverses
。
Definition Subring
A subset S of a ring R is a subring of R if S is itself a ring with the operations of R.
。
Theorem 12.3 Subring Test
A nonempty subset S of a ring R is a subring if S is closed under subtraction and multiplication—that is, if a-b∈S and ab∈S whenever a,b∈S.
.
Subring的unity不一定等于ring的unity
------------------------------------------------------
13  Integral Domains
动机：
To a certain degree, the notion of a ring was invented in an attempt to put the algebraic properties of the integers into an abstract setting(为什么要研究ring). 
A ring is not the appropriate abstraction of the integers, however, for too much is lost in the process. Besides the two obvious properties of commutativity and existence of a unity, there is one other essential feature of the integers that rings in general do not enjoy—the cancellation property.In this chapter, we introduce integral domains—a particular class of rings that have all three of these properties. Integral domains play a prominent role in number theory and algebraic geometry.（为什么要研究 Integral Domain）
.
Definition Zero-Divisors
A #zero-divisor# is a nonzero element a of a commutative ring R such that there is a nonzero element b∈R with ab=0.
Definition Integral Domain
An #integral domain# is a commutative ring with unity and no zero-divisors
.
Integral Domain之所以重要，是因为满足消去率
。
Theorem 13.1 Cancellation
Let a, b, and c belong to an integral domain. If a!=0 and ab=ac, then b=c.
.
field(+every unit)   ⊂   Integral Domain(+one unity+交换率+消去率)   ⊂   ring
（如果为了研究Z而定义了Integral Domain；那么为了研究Q，定义了field。）
.
Definition Field
A field is a commutative ring with unity in which every nonzero element is a unit.
。
a field can be thought of as simply an algebraic system that is closed under +,-,*,/(except by 0)
.
Theorem 13.2 Finite Integral Domains Are Fields
A finite integral domain is a field.
.
Definition Characteristic of a Ring
The #characteristic# of a ring R is the least positive integer n such that nx=0 for all x in R. If no such integer exists, we say that R has characteristic 0. The characteristic of R is denoted by char(R).
.
An infinite ring can have a nonzero characteristic.
.
Theorem 13.3 Characteristic of a Ring with Unity(对于有unity的ring，如何求char(R))
Let R be a ring with unity 1. If 1 has infinite order under addition, then the characteristic of R is 0. If 1 has order n under addition, then the characteristic of R is n.
.
Theorem 13.4 Characteristic of an Integral Domain
The characteristic of an integral domain is 0 or prime.
-------------------------------------------------------------------------
14 Ideals and Factor Rings
Definition Ideal
A subring 'A' of a ring R is called a (two-sided) #ideal# of R if for every r∈R and every a∈A both ra, ar∈A.(注意，ra,ar只针对算符*,即group的算符)
A◁R  ?
。
ideal   ⊂   subring
。
An ideal A of R is called a proper ideal of R if A is a proper subset of R
。
Theorem 14.1 Ideal Test
A nonempty subset A of a ring R is an ideal of R if
1. a-b∈A whenever a, b∈A.(类似subring test)
2. ra,ar∈A whenever a∈A and r∈R.
。
For any ring R, {0} and R are ideals of R. The ideal {0} is called the #trivial ideal#.
。
Let R be a commutative ring with unity and let a∈R.
The set <a>= {ra | r∈R} is an ideal of R called the #principal ideal generated by a#.
.
Let R be a commutative ring with unity and let a1,a2, . . . , an belong to R. Then I=<a1, a2, . . . , an>={r1a1+r2a2+...+rnan | ri∈R} is an ideal of R called the #ideal generated by a1,a2, . . . , an#.
.
Theorem 14.2 Existence of Factor Rings
Let R be a ring and let 'A' be a subring of R. The set of cosets {r+A |r∈R} is a ring under the operations (s+A)+(t+A) =s +t+A and (s+A)(t+A)=st+A  iff  A is an ideal of R.
.
Examples 11 and 12 illustrate one of the most important applications of factor rings—the construction of rings with highly desirable properties.
.
Definition Prime Ideal, Maximal Ideal
A #prime ideal# 'A' of a commutative ring R is a proper ideal of R such
that a, b∈R and ab∈A imply a∈A or b∈A. 
A #maximal ideal# 'A' of a commutative ring R is a proper ideal of R such that, whenever B is an ideal of R and A⊆B⊆R, then B=A or B=R.
定义prime ideal的动机来自整数
。
如何判断一个ideal is prime or maximal.
Theorem 14.3 R/A Is an Integral Domain If and Only If A Is Prime
Let R be a commutative ring with unity and let A be an ideal of R. Then R/A is an integral domain if and only if A is prime.
Theorem 14.4 R/A Is a Field If and Only If A Is Maximal
Let R be a commutative ring with unity and let A be an ideal of R. Then R/A is a field if and only if A is maximal.
。
Emmy Noether： whenever there is a symmetry in nature, there is also a conservation law,
--------------------------------------------------------
15 Ring Homomorphisms
Definitions Ring Homomorphism, Ring Isomorphism
A #ring homomorphism# f from a ring R to a ring S is a mapping from R to S that preserves the two ring operations; that is, for all a,b∈R,
f(a + b) =f(a) +f(b)    and    f(ab) =f(a)f(b).
A ring homomorphism that is both one-to-one and onto is called a #ring isomorphism#.
。
The correspondence f: x-->5x from Z4 to Z10 is a ring homomorphism.
f(x +y)=f(x) + f(y),左边的x+y是对5取模，右边的f(x) + f(y)是对10取模
。
Theorem 15.1 Properties of Ring Homomorphisms
Let f be a ring homomorphism from a ring R to a ring S. Let A be a subring of R and let B be an ideal of S.
1. For any r [ R and any positive integer n, f(nr) 5 nf(r) and
f(r^n )=(f(r))^n .
2. f(A)={f(a) | a∈A} is a subring of S.
3. If A is an ideal and f is onto S, then f(A) is an ideal.
4. f^-1(B) ={r∈R | f(r)∈B} is an ideal of R.
5. If R is commutative, then f(R) is commutative.
6. If R has a unity 1, S!={0}, and f is onto, then f(1) is the unity of S.
7. f is an isomorphism if and only if f is onto and Ker(f )={r ∈R | f(r) =0}={0}.
8. If f is an isomorphism from R onto S, then f^-1 is an isomorphism from S onto R.
.
Theorem 15.2 Kernels Are Ideals
Let f be a ring homomorphism from a ring R to a ring S. Then Ker(f)5 {r∈R | f(r)=0} is an ideal of R.
.
Theorem 15.3 First Isomorphism Theorem for Rings(Fundamental Theorem of Ring Homomorphisms)
Let f be a ring homomorphism from R to S. Then the mapping from R/Ker(f) to f(R), given by r+Ker(f)-->f(r), is an isomorphism. In symbols, R/Ker(f )≈f(R).
.
Theorem 15.4 Ideals Are Kernels
Every ideal of a ring R is the kernel of a ring homomorphism of R. In particular, an ideal A is the kernel of the mapping r --> r+A from R to R/A.
(The homomorphism from R to R/A given in Theorem 15.4 is called the #natural homomorphism# from R to R/A)
.
Theorem 15.5 Homomorphism from Z to a Ring with Unity
Let R be a ring with unity 1. The mapping f: Z S R given by n -->n*1 is a ring homomorphism.
.
Corollary 1 A Ring with Unity Contains Z n or Z
If R is a ring with unity and the characteristic of R is n>0, then R contains a subring isomorphic to Z n . If the characteristic of R is 0, then R contains a subring isomorphic to Z.
.
Corollary 2 Z m Is a Homomorphic Image of Z
For any positive integer m, the mapping of f: Z-->Zm given by x-->x mod m is a ring homomorphism
.
Corollary 3 A Field Contains Zp or Q (Steinitz, 1910)
If F is a field of characteristic p, then F contains a subfield isomorphic to Zp . If F is a field of characteristic 0, then F contains a subfield isomorphic to the rational numbers.
Since the intersection of all subfields of a field is itself a subfield (Exercise 11), every field has a smallest subfield (that is, a subfield that is contained in every subfield). This subfield is called the prime subfield of the field. It follows from Corollary 3 that the prime subfield of a field of characteristic p is isomorphic to Zp , whereas the prime subfield of a field of characteristic 0 is isomorphic to Q.
。
Theorem 15.6 Field of Quotients
Let D be an integral domain. Then there exists a field F (called the #field of quotients# of D) that contains a subring isomorphic to D.
.
---------------------------------------------------------------------
16 Polynomial Rings
Definition Ring of Polynomials over R
Let R be a commutative ring. The set of formal symbols
R[x] ={an*x^n + an-1*x^(n-1)+... + a1x1+a0 | ai∈R,  n is a nonnegative integer}
is called the ring of polynomials over R in the indeterminate x.
Two elements
an*x^n + an-1*x^(n-1)+... + a1x1+a0     and
bn*x^n + bn-1*x^(n-1)+... + b1x1+b0
of R[x] are considered equal if and only if ai=bi for all nonnegative integers i. (Define ai= 0 when i>n and bi=0 when i>m.)
.
deg( f(x)) = n
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17  Factorization   of Polynomials
-------------------------------------------------------------
18 Divisibility in Integral Domains
Definition Associates, Irreducibles, Primes
Relating the definitions above to the integers may seem a bit confus-
ing. The source of the confusion is that in the case of the integers,
the concepts of irreducibles and primes are equivalent, but in general, as
we will soon see, they are not.
.
Theorem 18.1 Prime Implies Irreducible
Theorem 18.2 PID Implies Irreducible Equals Prime
==================================
4   Field
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19 Vector Spaces
Definition Vector Space
A set V is said to be a vector space over a field F if V is an Abelian group under addition (denoted by 1) and, if for each a∈F and v∈V, there is an element av in V such that the following conditions hold for all a, b in F and all u, v in V.
1.a(v + u) = av + au
2.(a + b)v = av + bv
3.a(bv) = (ab)v
4.1v = v
.
Definition Subspace
Let V be a vector space over a field F and let U be a subset of V. We
say that U is a subspace of V if U is also a vector space over F under
the operations of V
-----------------------------------------------------------------
20 Extension Fields
Definition Extension Field
A field E is an #extension field# of a field F if F⊆E and the operations of F are those of E restricted to F.
.
Theorem 20.1 Fundamental Theorem of Field Theory
(Kronecker’s Theorem, 1887 )
Let F be a field and let f(x) be a nonconstant polynomial in F[x]. Then there is an extension field E of F in which f (x) has a zero.
.
EXAMPLE 1 Let f(x) =x^2+1∈Q[x]. Then, viewing f (x) as an element of E[x]= (Q[x]/<x^2 +1>)[x], we have
f (x+<x^2+1>) = ...= 0+<x^2+1>
the point we wish to emphasize here is that we have constructed a field that contains the rational numbers and #a zero for the polynomial x^2+1 #by using only the rational numbers. No knowledge of complex numbers is necessary. Our method utilizes only the field we are given.
（也就是说存在一个x^2+1的零元（不是sqrt(-1)）,可能我们现在还不清楚这个零元到底是什么(所以暂时发明了sqrt(-1)使得逻辑推理能够继续下去)）
。
Definition Derivative 多项式导数
Let f(x)= an x^n+an-1 x^(n-1)+...+a1x+a0 belong to F[x]. The derivative of f(x), denoted by f9(x), is the polynomial n*an*x^(n-1)+(n-1)an-1*x^(n-2) +...+a1 in F[x].
.
Lemma Properties of the Derivative 由多项式导数定义可以推导出多项式求导法则
Let f(x) and g(x) [ F[x] and let a [ F. Then
1. (f(x)+g(x))'= f'(x) +g'(x).
2. (af(x))' = af '(x).
3. (f(x)g(x))'= f'(x)g(x)+g'(x)f(x)
-----------------------------------------------------
21 Algebraic Extensions
Definition Types of Extensions
Let E be an extension field of a field F and let a∈E. 
We call a #algebraic代数数# over F if a is the zero of some nonzero polynomial in F[x]. 
If a is not algebraic over F, it is called #transcendental超越数#over F. 
An extension E of F is called an #algebraic extension# of F if #every# element of E is algebraic over F. 
If E is not an algebraic extension of F, it is called a #transcendental extension# of F. 
An extension of F of the form F(a) is called a #simple extension# of F.
.
F(x) is the field of quotients of F[x]:
F(x)={ f(x)/g(x) | f(x), g(x)∈F[x], g(x) !=0}.
.
Theorem 21.1 Characterization of Extensions(为什么要区分代数数和超越数)
Let E be an extension field of the field F and let a∈E. 
If a is transcendental over F, then F(a)≈F(x). 
If a is algebraic over F, then F(a)≈F[x]/<p(x)>, where p(x) is a polynomial in F[x] of minimum degree such that p(a)=0. Moreover, p(x) is irreducible over F.
。
If E is an extension field of F, we may view E as a vector space over F
。
Definition Degree of an Extension
Let E be an extension field of a field F. We say that E has degree n over F and write [E:F]= n if E has dimension n as a vector space over F. 
If [E:F] is finite, E is called a #finite extension# of F; otherwise, we say that E is an #infinite extension# of F.
.
Theorem 21.4 Finite Implies Algebraic
If E is a finite extension of F, then E is an algebraic extension of F.
.
Properties of Algebraic Extensions
Theorem 21.7 Algebraic over Algebraic Is Algebraic
If K is an algebraic extension of E and E is an algebraic extension of F, then K is an algebraic extension of F.
.
Corollary Subfield of Algebraic Elements
Let E be an extension field of the field F. Then the set of all elements of E that are algebraic over F is a subfield of E.
For any extension E of a field F, the subfield of E of the elements that are algebraic over F is called the #algebraic closure# of F in E
.
A field that has no proper algebraic extension is called #algebraically closed#
。
C is algebraically closed –--Fundamental Theorem of Classical Algebra
------------------------------------------------------------
22 Finite Fields（前几章内容的综合）
the most beautiful and important areas of abstract algebra—finite fields
。
Theorem 22.1 Classification of Finite Fields
For each prime p and each positive integer n, there is, up to isomorphism, a unique finite field of order p^n . 把这个finit field记为GF(p^n) (Galois field of order p^n)
。
Structure of Finite Fields
Theorem 22.2 Structure of Finite Fields
As a group under addition, GF(p^n ) is isomorphic to Zp⊕Zp⊕...⊕Zp   (n factors),
As a group under multiplication, the set of nonzero elements of GF(p^n) is isomorphic to Z ($_{p^n-1}$) (and is, therefore, cyclic).
.
Corollary 1  [GF(p^n ):GF(p)] =n
Corollary 2 GF(p n ) Contains an Element of Degree n
Let 'a' be a generator of the group of nonzero elements of GF( p^n ) under multiplication. Then 'a' is algebraic over GF(p) of degree n.
.
Theorem 22.3 Subfields of a Finite Field
For each divisor m of n, GF( p^n ) has a unique subfield of order p^m .Moreover, these are the only subfields of GF( p^n ).
theorems 22.2 and 22.3, together with Theorem 4.3, make the task of finding the subfields of a finite field a simple exercise in arithmetic.
---------------------------------------------------------------
23 Geometric Constructions
.
we call a real number a constructible if, by means of an unmarked straightedge, a compass, and a line segment of length 1, we can construct a line segment of length |a| in a finite number of steps.
.
尺规作图，化圆为方，三等分角为什么不可能
We now come to the crucial question. Starting with points in the plane of some field F, which points in the real plane can be obtained with an unmarked straightedge and a compass?
----------------------------------------------------------
24 Sylow Theorems
.
Definition Conjugacy Class of a
Let a,b∈G. We say that a and b are #conjugate# in G (and call b a conjugate of a) if xax^-1=b for some x in G. The conjugacy class of a is the set cl(a) ={xax^-1 | x∈G}.
.
---------------------------------------------------------
25 Finite Simple Groups
Definition Simple Group
A group is #simple# if its only normal subgroups are the identity subgroup and the group itself.
-----------------------------------------------------
27 Symmetry Groups
.
Definition Isometry
An #isometry# of n-dimensional space Rn is a function from Rn onto Rn that preserves distance.
.
Definition Symmetry Group of a Figure in R n
Let F be a set of points in Rn . The #symmetry group# of F in Rn is the set of all isometries of Rn that carry F onto itself. The group operation is #function composition#.
.
Theorem 27.1 Finite Symmetry Groups in the Plane
The only finite plane symmetry groups are Zn and Dn .
。
Classification of Finite Groups of Rotations in R3
Theorem 27.2 Finite Groups of Rotations in R 3
Up to isomorphism, the finite groups of rotations in R3 are Zn, Dn, A4, S4, and A5.
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The Alternating Group A4 of Even Permutations of {1, 2, 3, 4}
The Alternating Group A5 of Even Permutations of {1, 2, 3, 4, 5}
S4 is the symmetric group of degree 4
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29   Symmetry   and Counting
In general, we say that two designs (arrangements of beads) A and B are equivalent under a group G of permutations of the arrangements if there is an element f in G such that f(A)= B. That is, two designs are equivalent under G if they are in the same orbit of G. the number of nonequivalent designs under G is simply the number of orbits of designs under G
we say that G #acts on# S if there is a homomorphism r from G to sym(S), the group of all permutations on S. (The homomorphism r is sometimes called the #group action#.)
denote the image of g under r as r($_g$)
x,y∈S are viewed as equivalent under the r   iff  r($_g$)(x)=y for some g in G.
Notice that when r is one-to-one, the elements of G may be regarded as permutations on S.
On the other hand, 
when r is not one-to-one, the elements of G may still be regarded as permutations on S, but there are distinct elements g and h in G such that r($_g$) and r($_h$) induce the same permutation on S [that is, r($_g$) (x)=r($_h$)(x) for all x in S]. Thus, a group acting on a set is a natural generalization of the permutation group concept.
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33 Cyclotomic Extensions
It was this discovery of the constructibility of the 17-sided regular polygon that induced Gauss to dedicate his life to the study of mathematics.
Manjul Bhargava
He not only broke new ground in that area but also discovered 13 more composition laws and developed a coherent mathematical framework to explain them.What made Bhargava’s work especially remarkable is that he was able to explain all his revolutionary ideas using only elementary mathematics.
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