高等代数

高等代数

基本Notation

Cardinality of X: \(\vert X \vert\) 这个set里有多少个元素

\[\mathbb{Z} = \{ \cdots, -2, -1, 0, 1, 2, \cdots \}\] \[\mathbb{Z}_{\ge 0} = \{ \cdots, -2, -1, 0, 1, 2, \cdots \}\] \[\mathbb{N} = \{ 1, 2, \cdots \}\] \[\mathbb{Q} = \{ {\frac{m}{n} | m,n \in \mathbb{Z} } \}\] \[M_n( \mathbb{Z}) := \text{ all } n \times n \text{ matrices with integer entries}\]

general linear group of degree n

\[GL_n(R) := \{ A \in M_n(\mathbb{R}) \vert A \text{ is invertible} \} \\ (GL_n( \mathbb{R}), \times) \text{ is a group}\\\] \[SL_n( \mathbb{Z}) = \{ A \in M_n ( \mathbb{Z}) \vert det(A) = 1 \}\]

Definition Binary operation

Definition

Homomorphism

A map \(\phi : G_1 \to G_2\) is a group homomorphism if \(\phi\) preserves the operations:

\[\phi( g \ast_1 g') = \phi(g) \ast_2 \phi(g') \,\, , \forall g, g' \in G_1\]

所以 homomorphism 有时候更强调的是一个map, 不是强调两个group的关系。

Isomorphism

Isomorphism 只是 Homomorphism 的一种: 一种bijection 的homomorphism。

这时候我们可以强调这两个group 的关系. \(G_1\) and \(G_2\) is isomorphic 。

denoted as:

\[\phi : G_1 \simeq G_2 \text{ or } G_1 \cong G_2\]

Automorphism

也是强调一个 map：是一个 Isomorphism onto itself.

Symmetric Group

我们先定义一个general 的function, for a non empty set X.

\[sym(X) := \{ f: X \to X | \text{ f is a bijective map }\}\]

如果 \(X = \{ 1,2,...,n \}\) 有限集, 那么我们就定义了 Symmetric Group

\[S_n = sym(X)\]

就是一个 Permutation, \(X\) 的不同的排列方式。

Disjoint cycles

然后我们可以用 Disjoint Cycles 来表示她:

\(\sigma = \begin{align*} \left(\begin{array}{ccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 2 & 5 & 1 & 4 & 7 & 6 & 8 \\ \end{array} \right) \end{align*} = (1,3,5,4) (6,7) (2) (8) = (1,3,5,4) (6,7)\)

Disjoint Cycle Product theorem

Any non-identity permutation of \(S_n\) can be uniquely expressed (up to the ordering of cycles) as a product of disjoint cycles, where each cycle is of length at least 2.

Decomposition into transpositions theorem

\(S_n = \langle (1,2),(2,3),(3,4),....(n-1,n) \rangle\)

Signature (Even/ Old permutation)

\[sgn(\sigma) = \sigma(D) = \prod_{ 1 \le i < j \le n} (x_{\sigma(i)} - x_{\sigma(j)} )\]

        If \(sgn(\sigma) = 1\), \(\sigma\) is an even permutation.
        If \(sgn(\sigma) = -1\), \(\sigma\) is an odd permutation.

Alternating group \(A_n\)

\[A_n := \{ \sigma \in S_n | \sigma \text{ is an even permutation }\}\]

  note: \((A_n, \circ)\) is a subgroup of \((S_n, \circ)\) ,

3-cycles generating \(A_n\) theorem

: \(A_n\) is generated by 3-cycles.

\(\vert S_n \vert = 2 \vert A_n \vert\)

\[(1,2) A_n := \text{ all odd permutations in } S_n\]

所以这是一个 disjoin union 关系。就是两个 disjoin的 set 的union 的意思。

\[S_n = A_n \coprod (1,2) A_n.\]

Subsets product on a group

\((G, \ast)\) is a group, \(R, S, T \subset G\)

If \(S = \{ g \}\)
\(gT = \{gt |t \in T\}\)

\[ST := \{ st| s \in S, t \in T \}\]

Also We have \((RS)T = R(ST)\)

Arthur Cayley’s Theorem

每一个 finite group 都是一个 subgroup of \(S_n\) where \(n:= \vert G \vert\)

点我展开证明proof

其实就是要找到一个bijection: \(f : G \to S_n\)

这个证明其实比想象中的简单：我们先把每个元素给它分一个不重复的index : \(G = \{ g_1, g_2, ..., g_n \}\) 然后我们知道 {1,2,…,n} 必然是在 \(\in S_n\) 里的。

Euler’s phi-function

\[\varphi(n) = \vert \{ s \in \mathbb{N}: s \le n, gcd(s,n) = 1 \}\]

即, n 有多少个不能整除它的数。

几个重要的结论：

•     \(p\) be a prime number: \(\varphi(p^s) = p^{s-1} (p - 1)\) for \(s \in \mathbb{N}\)

•     If \(r,s \in \mathbb{N}\) are coprime. \(\varphi(rs) = \varphi(r) \varphi(s)\)

•     For integer \(n \ge 2\) let \(n= p_1^{s_1} \cdots p_s^{s_s}\), then \(\vert U ( \mathbb{Z} / (n)) \vert = \varphi(n) = \varphi(p_1^{s_1}) \cdots p_k^{s_k}\)

点我展开证明proof

1. 直观理解的话, 如果把 \(p^s\) 分成 s 份, \(\{1,2,...,p\}, \{p +1, p+2, ..., 2*p \}, ...\) (一共 \(p^{s - 1}\) 份) 每一份其实都有且恰好只有一个 p 可以 divide \(p^s\), 所以每个数减掉就是 \(\varphi(n) = p^s - p^{s-1} = p^{s-1} ( p -1)\)

2.

Euler’s Theorem

For all \(a \in \mathbb{Z}\) coprime to \(n\) :

\[a^{\varphi(n)} \equiv 1 \text{ (mod n)}\]

点我展开证明proof

如果用之前的结论, 证明起来还是很方便的：

首先： \(a_{}^{}\) coprime to \(n_{}^{}\) , then \([a]_n ( \text{ also read as } \bar{a}) \in U( \mathbb{Z} / (n))\) the multiplicative group.

Index of a subgroup

Let \(G_{}^{}\) be a group and \(H \le G\) (a subgroup). The index of \(H \text{ in } G\) is:

\[\vert G \vert = \vert G: H \vert \vert H \vert\]

is the number of left (or right) cosets of H in G.

Normal subgroup

\[N \vartriangleleft G \Leftrightarrow \text{ if } g n g^{-1} \in N, \forall n \in N, g \in G.\]

Normalizer

Let \(H \le G \text{ and } g \in G\)

      \(g_{}^{}\) normalizes \(H_{}^{}\) if \(g H g^{-1} = H\)

\[N_G (H) := \{ g \in G | gH g^{-1} = H \}\]

            is the Normalizer of \(H_{}^{}\) in \(G_{}^{}\)

Set of left cosets

\[\bar{g} := gH\]

Denote

\[G/H := \{ \bar{g} | g \in G \}\]

is the set of all left cosets of H in G.

where

Direct product of groups

\[G = G_1 \times G_2 = \{ (x_1, x_2) | x_i \in G_i \}\]

Or generally:

\[G = \prod_{i}^{s} G_i = G_1 \times \cdots \times G_s\]

When \((G_i, +)\) are abelian, we use

\[\oplus_{i=1}^{s} G_i\]

Fundamental theorem of finitely generated abelian group

Let \(G_{}^{}\) be a finitely generated abelian group. Then

\[\begin{align*} G &= F \times T \\ \\ T &:= \{ g \in G| o(g) < \infty \} \text{ called torsion subgroup of G}\\ \\ F &\simeq \mathbb{Z}/(p_1^{n_1}) \oplus \cdots \oplus \mathbb{Z} / (p_r^{n_r}) \text{ called free abelian group of rank } s\\ & \text{ for some primes } p_i , \text{ where it is possible to have } p_i = p_j \text{ for } i \neq j \end{align*}\]

Solvable group

If a group \(G_{}^{}\) is solvable if it has a subnormal series \(1 = H_0 \vartriangleleft H_1 \vartriangleleft \cdots \vartriangleleft H_n = G\) such that all the factors \(H_{i+1} / H_i\) are abelian.