The Alternating Group

The Alternating group

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 8 December 2010

The alternating group An

Definition. The Alternating group An is the subgroup of even permutations of Sn.

The alternating group An is the kernel of the sign homomorphism of the symmetric group; An= kerϵ, where ϵ: Sn ±1 σ detσ.

HW: Show that An is a normal subgroup of Sn.

HW: Show that An=n!/2.

Conjugacy classes

Since An is a normal subgroup of Sn, An is a union of conjugacy classes of Sn. Let 𝒞λ be a conjugacy class of Sn corresponding to a partition λ= λ1, λ2, , λk Then the following proposition says:
  1. The conjugacy class 𝒞λ is contained in An if an even number of the λi are even numbers.
  2. If the parts λi of λ are all odd and are all distinct then 𝒞λ is a union of two conjugacy classes of An and these two conjugacy classes have the same size.
  3. Otherwise 𝒞λ is also a conjugacy class of An.

Suppose that σAn. Let 𝒞λ denote the conjugacy class of σ in An and let 𝒜σ denote the conjugacy class of σ in An.

  1. Then σ has an even number of cycles of even length.
  2. 𝒜σ = 𝒞σ 2 , if all cycles of σ are of different odd lengths; 𝒞σ, otherwise.

The proof of Proposition 2.1 uses the following lemma.

Let σAn, and let λ= λ1, λ2, , λk be the cycle type of σ. Let γλ be the permutation given, in cycle, notation by γλ= 1,2,,λ1 λ1+1, λ1+2, , λ1+λ2+1, . Let Sσ denote the stabilizer of σ under the action of Sn on itself by conjugation. Then,

  1. SσAn if and only if SγλAn
  2. SγλAn if and only if γλ has all odd cycles of different lengths.

An is simple, n4.

A group is simple if it has no nontrivial normal subgroups. The trivial normal subgroups are the whole group and the subgroup containing only the identity element.

  1. If n4 then An is simple.
  2. The alternating group A4 has a single nontirival normal subgroup given by N= 1234, 2143 , 3412 , 4321 , where the permutations are represented in one-line notation.

The proof of Theorem 1.4 uses the following lemma.

Suppose N is a normal subgroup of An, n>4, and N contains a 3-cycle. Then N=An.


[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

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