浅溜一版裙论。
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.
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.
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 \(gh^{-1}ahg^{-1}\in a\).
Center of group \(Z(G)=C_G(G)\).
Normalizer \(N_G(A)\) contains elements with \(gA=Ag\).
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 \(B\). Proved by projection \(A\to AB/B\). Proof of this theorem suggests a way of prove \(A/B\cong C/D\) 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 \(AB\geq A\text{ as well as }B\geq A\cap B\).
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.
Fourth isomorphism theorem: subgroups containing normal subgroup corresponds with subgroup of quotient group.
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.
标签：Phi,group,varphi,优雅,专利,裙论,each,homomorphism,subgroup From： https://www.cnblogs.com/Troverld/p/18125920