Difference between revisions of "Alternating group"
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The '''alternating group''' on a [[finite]] [[set]] <math>M</math> is the [[group]] of [[symmetric group | even permutations]] on the set <math>M</math>; it is denoted <math>A_M</math>, <math>\mathfrak{A}_M</math>, or <math>\text{Alt}(M)</math>. When <math>M = \{1, \dotsc, n\}</math>, this group is denoted <math>A_n</math>, <math>\mathfrak{A}_n</math>, or <math>\text{Alt}(n)</math>. This is a [[normal subgroup]] of the [[symmetric group]]; and for <math>n=3</math> or <math>n\ge 5</math>, it is in fact a [[simple group]]. | The '''alternating group''' on a [[finite]] [[set]] <math>M</math> is the [[group]] of [[symmetric group | even permutations]] on the set <math>M</math>; it is denoted <math>A_M</math>, <math>\mathfrak{A}_M</math>, or <math>\text{Alt}(M)</math>. When <math>M = \{1, \dotsc, n\}</math>, this group is denoted <math>A_n</math>, <math>\mathfrak{A}_n</math>, or <math>\text{Alt}(n)</math>. This is a [[normal subgroup]] of the [[symmetric group]]; and for <math>n=3</math> or <math>n\ge 5</math>, it is in fact a [[simple group]]. | ||
| − | <math>A_n</math> is also the group of determinant-preserving permutations of the rows of an <math>n \times n</math> [[matrix]]. | + | <math>A_n</math> is also the group of [[determinant]]-preserving permutations of the rows of an <math>n \times n</math> [[matrix]]. |
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Latest revision as of 10:47, 4 March 2022
The alternating group on a finite set 
is the group of even permutations on the set ; it is denoted 
, 
, or 
. When 
, this group is denoted 
, 
, or 
. This is a normal subgroup of the symmetric group; and for 
or 
, it is in fact a simple group.
is also the group of determinant-preserving permutations of the rows of an 
matrix.
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