数学上，一个置换群 是一个群
G
{\displaystyle G}
M
{\displaystyle M}
置换 ，
G
{\displaystyle G}
M
{\displaystyle M}
M
{\displaystyle M}
对称群
Sym
(
M
)
{\displaystyle {\text{Sym}}(M)}
对称群 的一个子群 。如果
M
{\displaystyle M}
n
{\displaystyle n}
M
{\displaystyle M}
S
n
{\displaystyle S_{n}}
置换群到被置换的元素的应用称为群作用 ；它在对称性和组合论以及数学的其他很多分支中有应用。
置换通常写作轮换形式，例如，在轮换指标 计算中，给定集合
M
=
{
1
,
2
,
3
,
4
}
{\displaystyle M=\{1,2,3,4\}}
M
{\displaystyle M}
g
{\displaystyle g}
g
(
1
)
=
2
,
g
(
2
)
=
4
,
g
(
4
)
=
1
{\displaystyle g(1)=2,g(2)=4,g(4)=1}
g
(
3
)
=
3
{\displaystyle g(3)=3}
(
1
,
2
,
4
)
(
3
)
{\displaystyle (1,2,4)(3)}
(
1
,
2
,
4
)
{\displaystyle (1,2,4)}
3
{\displaystyle 3}
(
1
2
4
)
{\displaystyle (1\ 2\ 4)}
常见的置换群 [ 编辑 ]
M
=
{
1
,
2
}
{\displaystyle M=\{1,2\}}
[ 编辑 ]
(
1
)
,
(
1
2
)
{\displaystyle (1),(1\ 2)}
M
=
{
1
,
2
,
3
}
{\displaystyle M=\{1,2,3\}}
[ 编辑 ]
(
1
)
,
(
1
2
)
,
(
1
3
)
,
(
2
3
)
,
(
1
2
3
)
,
(
1
3
2
)
{\displaystyle (1),(1\ 2),(1\ 3),(2\ 3),(1\ 2\ 3),(1\ 3\ 2)}
M
=
{
1
,
2
,
3
,
4
}
{\displaystyle M=\{1,2,3,4\}}
[ 编辑 ]
(
1
)
,
{\displaystyle (1),}
(
1
2
)
,
(
1
3
)
,
(
1
4
)
,
(
2
3
)
,
(
2
4
)
,
(
3
4
)
,
{\displaystyle (1\ 2),(1\ 3),(1\ 4),(2\ 3),(2\ 4),(3\ 4),}
(
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),(1\ 3\ 2),(1\ 2\ 4),(1\ 4\ 2),(1\ 3\ 4),(1\ 4\ 3),(2\ 3\ 4),(2\ 4\ 3),}
(
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
)
{\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)}
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" . 
