Basic Concepts

Lec1

Definition

A group is a set \(G\) with a binary operation \(*\):\((a,b) \mapsto a*b\), satisfying:

1. \((a*b)*c=a*(b*c).\)

2. \(\exist\) an element \(e \in G\), called the identity, \(s.t.e*a=a*e=a,\forall a \in G.\)

3. For any \(a\in G\),\(\exist\) \(a^{-1}\), called an inverse of \(a\), \(s.t.a*a^{-1}=a^{-1}*a=e\)

If \(a*b=b*a,\forall a,b\in G\),\(G\) is called abelian or commutative.

Example

① \((ℤ,+)\) is an abelian group.The identity is \(0\).

② \((V,+)\) is an abelian group.(\(V\) is vector space over \(F\))

③ \((ℤ/mℤ,+)\) is a group.The identity is \(\overline{0}\).

④ Let \((G,*)\) and \((H,\) ◦\()\) be groups.We define \((G\times H,\Delta):(g,h)\Delta(g',h')=(g*g',h\) ◦ \(h')\).It's a direct product of \(G\) and \(H\), which is also a group.

Properties

Let \(G\) be a group, 1. The identity \(e\) is unique.\(proof\): If \(e'\) is also an identity, then \(e=e*e'=e'\).

2. The inverse \(a^{-1}\) of \(a\) is unique. \(proof\): If \((a^{-1})'\) is also an inverse of \(a\), then \(a^{-1}=a^{-1}*e=a^{-1}*(a*(a^{-1})')=(a^{-1}*a)*(a^{-1})'=(a^{-1})'.\)

3. \((a^{-1})^{-1}=a.\)

4. \((a*b)^{-1}=b^{-1}*a^{-1}\). By induction, \((a_1*a_2*...*a_n)^{-1}=a_n^{-1}*a_{n-1}^{-1}*...*a_1^{-1}.\)

5. (Cancellation law) \(a*b=a*c \Rightarrow b=c\). \(b*a=c*a \Rightarrow b=c\).

Lec2

Definition

(1) \({s_1,...,s_n} \in G\) is a set of generators of \(G\) if every element of \(G\) can be written as a product of \(s_1,...,s_n\) and \(s_1^{-1},...s_n^{-1}.\)

(2) A relation is an equality consisting of the generators and their inverses.

Write \(G=\lang s_1,...,s_n|R_1,...,R_m\rang\) if every relation can be deduced from \(R_1,...R_m\).

Example

$\mathbb{Z}_n=\mathbb{Z}/n\mathbb{Z} \cong \lang x|x^n=e \rang $.

\(\mathbb{Z}_6 \cong \lang r,s|r^3=1,s^2=1,rs=sr\rang\).

Definition

Let \(\Omega\) be a set.The set \(S_\Omega=\) { bijections \(\sigma:\Omega \longrightarrow \Omega\) } has a \(gp\) struct: \(gp\) struct : \(\sigma\tau=\sigma\)◦\(\tau\) identity:\(e=id_\Omega\) inverse:inverse of map

Example

\(S_n=S_{1,2,...n},|S_n|=n!.\) Elements of \(S_n\): the first expression: \(\sigma= \begin{pmatrix} 1 & 2 & ... & n \\ \sigma_1 & \sigma_2 & ... & \sigma_n \end{pmatrix}\) , where \(\sigma_i=\sigma(i)\). the second expression: consider \(n=7\), \(\sigma= \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 7 & 5 & 1 & 3 & 2 & 6 & 4 \\ \end{pmatrix}\), we can also write as \(\sigma=(1 7 4 3)(2 5)(6)\), where \((a_1,...a_r)\) with \(a_1,...,a_r\) distinct is called a cycle,that maps \(a_1 \mapsto a_2 \mapsto ... \mapsto a_r \mapsto a_1\).

Notes

① Disjoint cycles commute with each other. \((a_1\) \(a_2)\) is called a transposition. ② \((a_1,a_2,...,a_r)=(a_2,a_3,...,a_1)\). \((a_1,...,a_r)^{-1}=(a_r,...,r_1).\) ③ \(S_1\) is trival. \(S_2=\) {\(1,(1\) \(2)\)} \(\cong \mathbb{Z}_2\) \(S_n\) is not abelian for all \(n \ge3.\)

Info

① \(S_n\) is generated by \((i\) \(j),1 \le i<j \le n\). ② \(S_n\) is generated by \((i,i+1)\), \(i=1,...n-1\). ③ \(S_n\) is generated by \((1\) \(2),(1\) \(2\) ... \(n)\).

Group isomorphisms(群同构)

Definition

Two groups \((G,*)\) and \((H,\) ◦\()\) are isomorphic denoted by \(G \cong H\),if there is a bijection \(\psi:G \longrightarrow H,\) s.t. \(\psi(g*g')=\psi(g)\) ◦ \(\psi(g'),\forall g,g' \in G.\)

Lec3

Definition

A group \(G\) is cyclic if \(\exist x \in G,s.t.G=\{x^k|k \in \mathbb{Z}\}\).

Two cases

\(|G|=n<\infty,G=\{e,x,x^2,...,x^{n-1} \},G \cong \mathbb{Z}_n\).

\(|G|=\infty,G \cong \mathbb{Z}.\)

Definition

设\(A\)是\(G\)的一个子集，则\(A\)生成的子群\(\lang A\rang:=\{a_1^{e_1}...a_r^{e_r}|r\geqslant0,a_i\in A,e_i=\pm 1 \}.\)

特别地，如果\(G\)是阿贝尔群，\(A=\{a_1,...,a_n\}\)，则\(\lang A \rang=\{a_1^{k_1}...a_n^{k_n}|k_i \in \mathbb{Z}\}.\)