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Difference between revisions of "Alternating group"

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Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012030/a01203019.png" /> is the non-Abelian simple group of smallest possible order.
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Note that $A_5$ is the non-Abelian simple group of smallest possible order.

Revision as of 19:25, 17 April 2014

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 $S_n$ 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.

References

[1] M. Hall, "Group theory" , Macmillan (1959)


Comments

Note that $A_5$ is the non-Abelian simple group of smallest possible order.

How to Cite This Entry:
Alternating group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alternating_group&oldid=31834
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article