## 整系数的分式线性变换群仅有9个有限子群

### 证明整系数的分式线性变换群的子群仅有九个。

$$D_n=\{e,a,a^2, \cdots, a^{n-1}, b, ab, a^2b, \cdots,a^{n-1}b\},$$

$$|D_n|=2|C_n|=2n,$$

$D_n$ 不是Abel群，因为 $ba=a^{n-1}b$。也就是说，$b^2=1$以及$b^{-1}ab =a^{-1}$。关于$D_n$的小群的结构，有 $D_1\cong C_2$。另外，$D_2$群是最小的非循环群。（这说明一二三阶群必然是循环群。）

$$\left[\begin{array}{cc} F_{n-1}& F_{n} \\ F_{n}& F_{n+1} \end{array}\right]$$

$$(x^n-1) / (x-1)=x^{n-1}+\cdots+ x^2+x+1$$

$$m^{(n)}(x)={\zeta}^nx +b({\zeta}^{(n-1)}+\cdots+ {\zeta}^2+\zeta+1)=x.$$

$$b^s=(xax^{-1})^s =x^s (a^s) x^{-s}= x^s x^{-s}=1,$$

## 定理的证明

1. Joseph A. Gallian, Contemporary Abstract Algebra, 5th ed., Houghton Mifﬂin, Boston, MA, 2002.
2. R. C. Lyndon and J. L. Ullman, Groups of elliptic linear fractional transformations, Proc. Amer. Math. Soc. 18 (1967), 1119–1124.
3. Dresden, Gregory P. “There are only nine finite groups of fractional linear transformations with integer coefficients.Mathematics Magazine (2004): 211-218.
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