Alternating group

The alternating group on a finite set $M$ is the group of even permutations on the set $M$; it is denoted $A_M$, $\mathfrak{A}_M$, or $\text{Alt}(M)$. When $M = \{1, \dotsc, n\}$, this group is denoted $A_n$, $\mathfrak{A}_n$, or $\text{Alt}(n)$. This is a normal subgroup of the symmetric group; and for $n=3$ or $n\ge 5$, it is in fact a simple group.

$A_n$ is also the group of determinant-preserving permutations of the rows of an $n \times n$ matrix.

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See also