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# 问求学习笔记-置换群与拉格朗日定理

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2019/03/09 Share

# 置换群(Permutation Group)

the binary operation is the composition of maps.

## Cycle Notation

A permutation $\sigma\in S_X$ is a cycle of length $k$ if there exist elements $a_1,a_2,…a_k\in X$ such that
$$\sigma(a_1) = a_2$$
$$\sigma(a_2) = a_3$$
$$…$$
$$\sigma(a_k) = a_1$$
and $\sigma(x)=x$ for all other elements $x\in X$.

We write $(a_1,a_2,…a_k)$ to denote the cycle $\sigma$.

Cycles are the building blocks of all permutations.循环是所有排列的基石。

Two cycles in $S_X$, $\sigma=(a_1,a_2,…a_k)$, $\tau=(b_1,b_2,…b_l)$, are disjoint if $a_i\neq b_j$ for all $i$ and $j$

Transpositions: a cycle of length 2 (任意两个数交换位置)

e.g. (253)=(23)(25)

# 反组(Dihedral Groups)

the nth dihedral group($D_n$): the group of rigid motions of a regular n-gon(正多边形的刚性运动=转动+反射)

$$r^n=1$$
$$s^2=1$$
$$srs=r^{-1}$$
($r,s$分别为转动和反射)

$$D_n={1,r,r^2,..,r^{n-1},s,sr,sr^2,…,sr^{n-1}}$$

# 陪集(Coset)

$G$为群，$H$为$G$的子群，定义

$1. g_1H=g_2H$;
$2. Hg_1^{-1}=Hg_2^{-1}$;
$3. g_1H\subset g_2H$;
$4. g_2\in g_1H$;
$5. g_1^{-1}g_2\in H$;

index of $H$: the number of left cosets of $H$ in $G$. 表示为$[G:H]$

# 拉格朗日定理

$$|G|=[G:H]|H|$$

# 费马与欧拉定理

Furthermore, for any integer $b$, $b^p\equiv b\pmod p$.