置换群

置换通常写作轮换形式，例如，在轮换指标计算中，给定集合${\displaystyle M=\{1,2,3,4\}}$ ，${\displaystyle M}$ 的一个置换${\displaystyle g}$ 若为${\displaystyle g(1)=2,g(2)=4,g(4)=1}$ 和${\displaystyle g(3)=3}$ ，可以写作${\displaystyle (1,2,4)(3)}$ ，或者更常见的写作${\displaystyle (1,2,4)}$ ，因为${\displaystyle 3}$ 保持不变；若对象有单个字母或数字表示，逗号也被省去，所以可以记作${\displaystyle (1\ 2\ 4)}$ 。

${\displaystyle M=\{1,2\}}$ 

${\displaystyle (1),(1\ 2)}$ 

${\displaystyle M=\{1,2,3\}}$ 

${\displaystyle (1),(1\ 2),(1\ 3),(2\ 3),(1\ 2\ 3),(1\ 3\ 2)}$ 

${\displaystyle M=\{1,2,3,4\}}$ 

${\displaystyle (1),}$  ${\displaystyle (1\ 2),(1\ 3),(1\ 4),(2\ 3),(2\ 4),(3\ 4),}$  ${\displaystyle (1\ 2\ 3),(1\ 3\ 2),(1\ 2\ 4),(1\ 4\ 2),(1\ 3\ 4),(1\ 4\ 3),(2\ 3\ 4),(2\ 4\ 3),}$  ${\displaystyle (1\ 2\ 3\ 4),(1\ 2\ 4\ 3),(1\ 3\ 2\ 4),(1\ 3\ 4\ 2),(1\ 4\ 2\ 3),(1\ 4\ 3\ 2),(1\ 2)(3\ 4),(1\ 3)(2\ 4),(1\ 4)(2\ 3)}$ 

• 群作用
• 本原群
• 置换
• 轮换
• 交错群

• John D. Dixon and Brian Mortimer. Permutation Groups. Number 163 in Graduate Texts in Mathematics. Springer-Verlag, 1996.
• Akos Seress. Permutation group algorithms. Cambridge Tracts in Mathematics, 152. Cambridge University Press, Cambridge, 2003.
• Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller and Peter M. Neumann. Notes on Infinite Permutation Groups. Number 1698 in Lecture Notes in Mathematics. Springer-Verlag, 1998.
• Alexander Hulpke. GAP Data Library "Transitive Permutation Groups" （页面存档备份，存于互联网档案馆）.

1. ^ 韩士安,林磊. 近世代数（第二版）. 北京: 科学出版社. 2009: 44. ISBN 9787030250612.