Alternating group

From Encyclopedia of Mathematics
Revision as of 17:18, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

of degree

The subgroup of the symmetric group consisting of all even permutations. is a normal subgroup in of index 2 and order . The permutations of , considered as permutations of the indices of variables , leave the alternating polynomial invariant, hence the term "alternating group" . The group may also be defined for infinite cardinal numbers , as the subgroup of consisting of all even permutations. If , the group is -fold transitive. For any , finite or infinite, except , this group is simple; this fact plays an important role in the theory of solvability of algebraic equations by radicals.


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


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

How to Cite This Entry:
Alternating group. Encyclopedia of Mathematics. URL:
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