裙论~不只是少女的专利，任何人都可以享受的优雅！

浅溜一版裙论。

O. Preliminaries

Binary operation is function $G\times G\to G$. Extra properties is optional:

• Associative: $(ab)c=a(bc)$.
• Commutative: $ab=ba$, in this case may call that the two elements commute in verbal form.

Equivalence $\sim$ is a binary relationship with properties:

• Reflexive $a\sim a$.
• Symmetric $a\sim b\Leftrightarrow b\sim a$.
• Transitive $a\sim b,b\sim c\Rightarrow a\sim c$.

I. Introduction to Groups

Dihedral Group $D_{2n}=\lang r,s\rang$。

Symmetric group $S_n$. Alternating group $A_n$ contains all even permutations. Permutation group is a subgroup of symmetric group.

Quaternion Group $Q_8=\{\pm1,\pm i,\pm j,\pm k\}$. $i^2=j^2=k^2=-1,ij=k,jk=i,ki=j$.

Klein $4$-group $V_4=\{1,i,j,k\}=Z_2\times Z_2$.

Group action. Element $g\in G$ has corresponding $\sigma_g:A\to A$, and $g\mapsto\sigma_g$ is homomorphism from $G$ to $S_A$. This homomorphism is called permutation representation.

Trivial homomorphism is the homomorphism that maps all elements to identity.

WHEN CHECKING HOMOMORPHISMS, ONLY NEED TO VERIFY THAT $\varphi(a)\varphi(b)=\varphi(ab)$, IDENTIDY IS INCLUDED IN THAT.

Trivial action is the action that all $\sigma_g=\iota$, the identity permutation. Another way to express it is $G$ act trivially over $A$.

Faithful action is the action with kernel being solely identity. Note that kernel of group action has no symbol notation, and it is exactly the same set as $\ker\varphi$, where $\varphi$ is permutation representation of the action.

II. Subgroups

Subgroup is set that is closed under products and inverses.

Subgroup criterion shows that subset is subgroup if and only if it is non-empty and $\forall x,y\in H$ there is $xy^{-1}\in H$. And, for finite groups, we only need to check that it is closed under multiplication, as finite group's elements all have finite order, after finite self-multiplications will gain inverse.

NOTE THAT WHEN CHECKING SUBGROUPS, FIRST CHECK THAT THE SUBGROUP IS NON-EMPTY (CHECK THE IDENTITY).

Centralizer of element $C_G(a)$ is the elements commute with $a$. Centralizer of subset is the elements that commute with every element from subset. Centralizer is group, as when $gag^{-1}=a$ and $hah^{-1}=a$ there must be $gh^{-1}ahg^{-1}=a$.

Center of group $Z(G)=C_G(G)$.

Normalizer $N_G(A)$ contains elements with $gA=Ag$.

$Z(G)\leq C_G(A)\leq N_G(A)\leq G$. When $A$ contains unique element, $C_G(a)=N_G(a)$.

Group action of $G$ on $S$, for element $s$ from $S$, stabilizer $G_s$ is the subset of $G$ with $g\cdot s=s$; kernel is intersection of all stabilizers.

Lattice: top-down describes all subgroups of a group, with $G$ at top and $\{1\}$ at bottom. A local minimal group that contains a subgroup appears over the subgroup and have an edge with it.

The subgroups of a subgroup act as nodes that can be reached by is, and quotient group isomorphic to the nodes above the particular subgroup.

III. Quotient Groups and Homomorphisms

整点抽象的。

第一同态定理说明，研究满同态就是在研究商群，研究商群就是在研究满同态；不满的同态取像集就得到了满的同态。

商群是群中将某些元素挤压在一块压成的东西：所有 $ab^{-1}\in H$ 的 $a,b$ 都被归入同一类。满同态同样有这样的忽略。

Fibers of a homomorphism is those elements mapped to a particular element, also known as coset of kernel. Representative of fiber or coset is any element from it.

$aH=bH\Leftrightarrow ab^{-1}\in H$, this property holds when $H\leq G$, not only when $H\unlhd G$.

$gng^{-1}$ is conjugate of $n$ by $g$, and $gNg^{-1}$ is conjugate of $N$ by $g$. $g$ normalizes $N$ when $gNg^{-1}=N$. Normal subgroup is normalized by every element. Normalizer of a group is those normalizes the group.

$\pi:G\to G/N$ is called natural projection or natural homomorphism.

Lagrange Theorem: The number of different left cosets of $N$ in $G$ is $\dfrac{|G|}{|N|}$. This is called index and denoted as $|G:N|$. For each divisor of order, there may not be a subgroup of such order; however in Abelian group there must.

Cauchy's Theorem: subgroup of prime divisor order exists.

$|HK|=\dfrac{|H||K|}{|H\cap K|}$.

$HK$ is subgroup if and only if $HK=KH$. This holds when one of $K,H$ is normal (in fact, one of them normalizes the other, which is $B\leq N_G(A)$, suffices), and when $K,H$ are both normal $KH$ should also be normal.

$A$ normalizes $B$ if $A$ is subset of normalizer, centralizes $B$ if is subset of centralizer.

First isomorphism theorem: $\ker\varphi\unlhd G$, $G/\ker\varphi\cong\Im(\varphi)$. Very useful when proving isomorphisms in forms of $A/B\cong C$.

Second isomorphism theorem: $AB/B\cong A/(A\cap B)$. Premise: $A$ normalizes $B$ ($A\leq N_G(B)$). Proved by projection $A\to AB/B$. Proof of this theorem suggests a way of prove $A/B\cong C/D$ type isomorphisms, which is simply view $C/D$ wholly as a group and use 1st Thm.. Also, 2nd Thm. itself can be a way to prove $A/B\cong C/D$ problems. However its form is complex in some way, thus not very flexible (?) This theorem is also called Diamond Theorem, mainly by the graph of $AB\geq A\text{ as well as }B\geq A\cap B$.

• Significance: Enlarge properties inside subgroup into a bigger subgroup. When $B$ and $A$ are complements (that is, $AB=G$ and $A\cap B=1$), properties of subgroup may transfered into properties of group.

Third isomorphism theorem: $(G/H)/(K/H)\cong G/K$. Proved also by projection $G/H\to G/K$. However 3rd Thm., as included subgroups embedding with each other, is also hard to use.

• Significance: Enlarge properties inside quotient group into properties of the original group.

Fourth isomorphism theorem: subgroups containing normal subgroup corresponds with subgroup of quotient group. Suggest primitive properties between quotient groups and original group. Drew in lattice diagram, this theorem is to say that the section 'above' a subgroup is 'isomorphic' to the lattice diagram of quotient group.

• Significance: Corresponding subgroups with quotient groups.

!!!!!

When we want to define homomorphism from quotient group $G/N$ to other groups, a common way to define it is to specify $\varphi(gN)$ for each $g$. However this definition need that $\varphi$ stays same through different representatives.

In fact, above definition can be seen as another homomorphism $\Phi$ from $G$ to the codomain group, each $\Phi(g)=\varphi(gN)$. In order that $\Phi(g)$ corresponding with valid $\varphi$, there should be $\Phi$ trivial on $N$, which is equivalent to that $N\leq\ker\Phi$. In fact, $\varphi$ is well-defined, if and only if $N\leq\ker\Phi$. A verbal way to express this, is to say that $\Phi$ factor throughs $N$; a diagram of this is, $G\xrightarrow{\pi}G/N\xrightarrow{\varphi}H$ and $G\xrightarrow\Phi H$, these two routes of projection generate same result for every element from $G$. Or, in other words, $\Phi=\varphi\circ\pi$.

Composition series $1=N_0\unlhd N_1\unlhd\dots\unlhd N_m=G$, with each $N_{i+1}/N_i$ simple. Each $N_{i+1}/N_i$ is called a composition factor.

Jordan-Hölder Theorem: finite group has unique composition series, rephrasing, different composition series have isomorphic composition factors.

Solvable group is group with $1=G_0\unlhd G_1\unlhd\dots\unlhd G_m=G$, each $G_{i+1}/G_i$ Abelian. Note that different from composition series, solvable group does not require each factor (?)(as the term cannot be called a factor) simple.

Finite solvable group should have each composition factor (note this case composition factor is that corresponded with composition series) prime order, as Abelian group can be dissected into factors, each factor a normal subgroup.

Phillip Hall (?) Theorem: finite group is solvable if and only if for each $n$ with $\gcd(n,\dfrac{|G|}n)=1$, there exists a subgroup of order $n$. (Which is, satisfied an extended version of Sylow's Theorem)

If $N$ and $G/N$ are solvable, then $G$ is solvable, by combining the series together through 3rd iso. Thm..

Transposition: $2$-cycle permutation.

IV. Group Action

Action on set has corresponding permutation representation. Reversely, any homomorphism from $G$ to symmetric group $S_A$ has corresponding group action, i.e. $g\cdot a=\varphi(g)(a)$. In fact, there exists bijection between group actions and symmetric group homomorphism.

Verbal form is that a group action affords or induces the associated permutation representation.

Orbit of $G$ containing $a$ is the elements of form $g\cdot a$. Orbits of $G$ act as partition of $G$. Transitive group action is action with only one orbit.

And, number of elements in an orbit containing $a$ is $|G:G_a|$, the index of stabilizer. In fact, there is bijection between left cosets of $G_a$ and elements of orbit, which is bijection $g\cdot a\mapsto gG_a$. This is conducted through the fact that, $g\cdot a=h\cdot a$ if and only if $gh^{-1}\in G_a$. This conclusion is called Orbit-Stabilizer Theorem.

The fact that each permutation has unique cycle decomposition can be deduced by analyzing orbits generated by $\lang\sigma\rang$ of permutation $\sigma$ over set $S$. Each orbit serves as a cycle from decomposition.

Group action on itself by left multiplication is to say that $G$ act on $G$ with $g\cdot a=ga$, left from group action and right from group binary operator. Action on itself by left multiplication is always transitive and faithful. The permutation representation induced by left multiplication on oneself is called left regular representation.

In fact, this action can be generalized to action by left multiplication on sets of elements, with $g\cdot aH=gaH$. This action is transitive, with stabilizer of left coset $1H$ be $H$, and kernel of action be $\bigcap\limits_{x\in G}xHx^{-1}$.

When $H$ is subgroup of $G$, left multiplication over left cosets has kernel be both normal subgroup of $G$ and subgroup of $H$ (therefore must be normal subgroup of $H$).

Cayley's Theorem: each group isomorphic to a permutation group.

Subgroup of index being smallest prime divisor of group's order is normal. Proved by showing that action by left multiplication over cosets over $H$ has kernel be $H$ itself.

Act on oneself by (left) conjugation is that $g\cdot a=gag^{-1}$, while right conjugation for that $g\cdot a=g^{-1}ag$. The two conjugation systems are equivalent.

$a,b$ conjugates in $G$ if $b=gag^{-1}$ for some $g$; that is, they are in the same orbit generated by conjugation, or conjugacy class in other words. Conjugation can also be generalized to $gSg^{-1}$ for subsets like $S$, also subsets can conjugate.

The stabilizer $G_S$ is those with $gSg^{-1}=S$, namely $N_G(S)$. And, when there is only one element in $S$, $N_G(s)=C_G(s)$. By Orbit-Stabilizer Theorem, number of conjugates of subset $S$ is $|G:N_G(S)|$, and of element $S$ is $|G:C_G(S)|$.

Class Equation: $|G|=|Z(G)|+\sum|G:C_G(r_i)|$, where $r_i$ is the representatives of distinct conjugacy classes of size at least $2$. This formula is actually very trivial, only dissect conjugacy classes into two parts, those with size $1$ represented as $|Z(G)|$, and those with size larger than $1$ represented by listing representatives in each conjugacy class.

Conjugation in matrix group $A\to PAP^{-1}$ is like change of basis; so as conjugation in symmetric group, with $\tau\sigma\tau^{-1}=(\tau(\sigma_1)\ \tau(\sigma_2)\dots\tau(\sigma_{i_1}))(\tau(\sigma_{i_1+1})\dots)\dots$, then there should be that two permutations conjugate if and only if they have the same cycle type, and number of conjugacy classes equals to integer partitions of $n$.

Normal group is the union of some conjugacy classes; that is, if $N\unlhd G$, then for every conjugacy class $K$, either $K\sube N$ or $N\cap K=\varnothing$. This is because, for $x\in N\cap K$, $gxg^{-1}\in N$ and $gxg^{-1}\in K$, thus $gxg^{-1}\in N\cap K$.

Simplicity of $A_5$ can be seen by analyzing conjugacy classes in $A_5$. CONJUGACY CLASSES IN $A_5$ IS NOT SIMILAR TO THAT OF $S_5$, AS EVEN PERMUTATIONS MAY CONJUGATED BY ODD PERMUTATIONS. IN FACT, EVEN IF SUBGROUPS CONTAINS A WHOLE CONJUGACY CLASS, THE CONJUGACY CLASS MAY BREAK INTO SMALLER PIECES IN SUBGROUP.

Automorphism is isomorphism with oneself. All automorphisms form a group, namely $\text{Aut}(G)$.

$G$ is able to act on normal subgroup $H$ by conjugation, as conjugation is closed under $H$. The action induces a permutation representation which is a homomorphism from $G$ to $\text{Aut}(H)$, with kernel be $C_G(H)$. Therefore, $G/C_G(H)$ is isomorphic with some subgroup of $\text{Aut}(H)$. Note this has premise of $H$ normal. And when $H$ is not normal, there is following:

Corollary: $N_G(H)/C_G(H)$ is isomorphic to some subgroup of $\text{Aut}(H)$, the one consists of all conjugation induced by elements of $N_G(H)$.

Conjugation by $g$ induces an automorphism over $G$, called inner automorphism. All such conjugation automorphisms form a group, namely $\text{Inn}(G)$, a subgroup of $\text{Aut}(G)$. There is $\text{Inn}(G)\cong G/Z(G)$.

A subgroup is characteristic if it conserves through all automorphisms, and is denoted as $H\operatorname{char} G$. Characteristic subgroup in priori should be normal, as conjugation is automorphism.

Automorphism group of cyclic group $Z_n$ is isomorphic to $(Z/nZ)^\times$, the reduced modulo group with order $\varphi(n)$.

1st Sylow Theorem: Sylow $p$-subgroup exists.

2nd Sylow Theorem: 1) Sylow $p$-subgroups conjugate. 2) $p$-subgroups of smaller order embeds in one of greater order.

3rd Sylow Theorem: 1) $n_p\equiv1\pmod p$; 2) $n_p=|G:N_G(P)|$, therefore $n_p\mid |G|$ hence $n_p\mid\dfrac{|G|}{p^\alpha}$.

For a Sylow $p$-subgroup, these following properties are equivalent:

• being unique Sylow $p$-subgroup.
• being normal.
• being characteristic.
• for $X$ which consists solely of $p$-power order elements, $\lang X\rang$ is $p$-group.

V. Direct and Semidirect Products and Abelian Groups

Direct product of groups gain a new group, each 'factor' of direct product is exactly called component or factor. Each component has an isomorphic image as normal subgroup of direct product, which indicates direct product is a way of 'combining' two groups, factors serves as normal subgroups.

Fundamental theorem of finite generated Abelian groups, is that each such group may be uniquely written as form $G=\Z^r\times Z_{n_1}\times Z_{n_2}\times\dots\times Z_{n_s}$, with restraints of $r\geq0,n_i>1,n_{i+1}\mid n_i$. $r$ is called free rank or Betti number, $n_1,\dots,n_s$ called invariant factors, and this decomposition called invariant factor decomposition of $G$.

Note that $Z_{mn}=Z_m\times Z_n$, when and only when $\gcd(n,m)=1$. Then each $Z_{n_i}$ can be continually factored into smaller pieces of coprime cyclic groups, and the full decomposition is called elementary divisor decomposition.

Commutator of $x,y$ is $[x,y]=x^{-1}y^{-1}xy$, with property $xy=yx[x,y]$. $[A,B]=\lang[a,b]\mid a\in A,b\in B\rang$, and $G'=\lang[x,y]\mid x,y\in G\rang$, namely commutator subgroup. I hate this notation, so $[G,G]$ instead. $[G,G]$ is in the union of every normal subgroup with quotient group Abelian. $H\unlhd G$ if and only if $[H,G]\leq H$.

Number of distinct way of written elements from $HK$ in form $hk$ is exactly $|H\cap K|$. Then there is that, if $H,K\unlhd G$ and $H\cap K=1$, then $HK\cong H\times K$, the former called internal direct product, latter external direct product. This theorem called recognition theorem (Reg. Thm.)

Semidirect product is originally defined over subgroups of $G$: when $H\unlhd G$ and $K\leq G$ and $H\cap K=1$, there could be $(h_1k_1)(h_2k_2)=h_1(k_1h_2k_1^{-1})k_1k_2=h_3k_3$, thus let $K$ act on $H$ by conjugation, which is $k\cdot h=k^{-1}hk^{-1}$, then $(h_1k_1)(h_2k_2)=(h_1(k_1\cdot h_2))(k_1k_2)$. Build bijection between $HK$ and $H\times K$, there is $(h_1,k_1)(h_2,k_2)=(h_1(k_1\cdot h_2),k_1k_2)$, this leads to definition of semidirect product without $G$:

When $K$ has a homomorphism $\varphi$ onto $\text{Aut}(H)$, then may define $(h_1,k_1)(h_2,k_2)=(h_1(k_1\cdot h_2),k_1k_2)$. This multiplication deduces a group $H\rtimes_\varphi K=:G$, with $H\cong\{(h,1)\}$ and $K\cong\{(1,k)\}$ thus able to identify $H,K$ as subgroups of $G$, thus $H\unlhd G$, $H\cap K=1$, $khk^{-1}=k\cdot h=\varphi(k)(h)$.

The following properties are equivalent:

• identity map between $H\rtimes K$ and $H\times K$ (this is valid as they are defined based on a same set) is homomorphism hence isomorphism.
• $K\unlhd G$.
• $\varphi$ is trivial homomorphism (mapping everything to identity)

Then we know that semidirect product is actually an extension of direct product.

Semidirect product is a neat way to classify groups of a certain order, according to the following property:

• Let $H\unlhd G,K\leq G$ and $H\cap K=1$. Consider $\varphi$, the left conjugation action by $K$ over $H$, is a homomorphism of $K$ to $\text{Aut}(H)$. Then there should be $HK\cong H\rtimes_\varphi K$. And when $|H||K|=|G|$, this is a way of dissect $G$ into smaller parts.

Such $K$ with $K\cap H=1$ and $G=HK$ is called component of $H$. Component of proper normal subgroups indicates a way of dissect, however not in every group such dissection may be found.

Conbine this property with Sylow Theorem, by which we may know the existence of Sylow $p$-groups (and probably the normality of some of them), and by that semidirect product generate group isomorphic with subgroup product $G$ may be classified.