Difference between revisions of "Alternating group"
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| − | The subgroup $A_n$ of the [[symmetric group]] $S_n$ consisting of all even permutations. $A_n$ is a normal subgroup in $S_n$ of index 2 and order $n!/2$. The permutations of $A_n$, considered as permutations of the indices of variables $x_1,\ldots,x_n$, leave the alternating polynomial $\prod(x_i-x_j)$ invariant, hence the term "alternating group". The group $A_m$ may also be defined for infinite cardinal numbers $m$, as the subgroup of $ | + | The subgroup $A_n$ of the [[symmetric group]] $S_n$ consisting of all even permutations. $A_n$ is a normal subgroup in $S_n$ of index 2 and order $n!/2$. The permutations of $A_n$, considered as permutations of the indices of variables $x_1,\ldots,x_n$, leave the alternating polynomial $\prod(x_i-x_j)$ invariant, hence the term "alternating group". The group $A_m$ may also be defined for infinite cardinal numbers $m$, as the subgroup of $S_m$ consisting of all even permutations. If $n>3$, the group $A_n$ is $(n-2)$-fold [[transitive group|transitive]]. For any $n$, finite or infinite, except $n=4$, this group is simple; this fact plays an important role in the theory of solvability of algebraic equations by radicals. |
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Latest revision as of 19:22, 4 April 2023
of degree $n$
The subgroup $A_n$ of the symmetric group $S_n$ consisting of all even permutations. $A_n$ is a normal subgroup in $S_n$ of index 2 and order $n!/2$. The permutations of $A_n$, considered as permutations of the indices of variables $x_1,\ldots,x_n$, leave the alternating polynomial $\prod(x_i-x_j)$ invariant, hence the term "alternating group". The group $A_m$ may also be defined for infinite cardinal numbers $m$, as the subgroup of $S_m$ consisting of all even permutations. If $n>3$, the group $A_n$ is $(n-2)$-fold transitive. For any $n$, finite or infinite, except $n=4$, this group is simple; this fact plays an important role in the theory of solvability of algebraic equations by radicals.
Comments
Note that $A_5$ is the non-Abelian simple group of smallest possible order.
References
- [1] M. Hall, "Group theory" , Macmillan (1959) Zbl 0084.02202
Alternating group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alternating_group&oldid=53576