# Alternating group

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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)