The Alternating Group

## The alternating group ${A}_{n}$

Definition. The Alternating group ${A}_{n}$ is the subgroup of even permutations of ${S}_{n}$.

The alternating group ${A}_{n}$ is the kernel of the sign homomorphism of the symmetric group; $An= kerϵ, where ϵ: Sn → ±1 σ ↦ detσ.$

HW: Show that ${A}_{n}$ is a normal subgroup of ${S}_{n}$.

HW: Show that $\left|{A}_{n}\right|=n!/2.$

## Conjugacy classes

Since ${A}_{n}$ is a normal subgroup of ${S}_{n}$, ${A}_{n}$ is a union of conjugacy classes of ${S}_{n}$. Let ${𝒞}_{\lambda }$ be a conjugacy class of ${S}_{n}$ corresponding to a partition $\lambda =\left({\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{k}\right)$ Then the following proposition says:
1. The conjugacy class ${𝒞}_{\lambda }$ is contained in ${A}_{n}$ if an even number of the ${\lambda }_{i}$ are even numbers.
2. If the parts ${\lambda }_{i}$ of $\lambda$ are all odd and are all distinct then ${𝒞}_{\lambda }$ is a union of two conjugacy classes of ${A}_{n}$ and these two conjugacy classes have the same size.
3. Otherwise ${𝒞}_{\lambda }$ is also a conjugacy class of ${A}_{n}$.

Suppose that $\sigma \in {A}_{n}$. Let ${𝒞}_{\lambda }$ denote the conjugacy class of $\sigma$ in ${A}_{n}$ and let ${𝒜}_{\sigma }$ denote the conjugacy class of $\sigma$ in ${A}_{n}$.

1. Then $\sigma$ has an even number of cycles of even length.

The proof of Proposition 2.1 uses the following lemma.

Let $\sigma \in {A}_{n}$, and let $\lambda =\left({\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{k}\right)$ be the cycle type of $\sigma$. Let γλ be the permutation given, in cycle, notation by $γλ= 1,2,…,λ1 λ1+1, λ1+2, …, λ1+λ2+1,⋯ ⋯.$ Let ${S}_{\sigma }$ denote the stabilizer of $\sigma$ under the action of ${S}_{n}$ on itself by conjugation. Then,

1. ${S}_{\sigma }\subset {A}_{n}$ if and only if ${S}_{{\gamma }_{\lambda }}\subset {A}_{n}$
2. ${S}_{{\gamma }_{\lambda }}\subset {A}_{n}$ if and only if ${\gamma }_{\lambda }$ has all odd cycles of different lengths.

## ${A}_{n}$ is simple, $n\ne 4$.

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 $n\ne 4$ then ${A}_{n}$ is simple.
2. The alternating group ${A}_{4}$ 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 ${A}_{n}$, $n>4$, and $N$ contains a $3$-cycle. Then $N={A}_{n}$.

## References

[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)