The Alternating group

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 8 December 2010

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; $$A_n=ker\left(ϵ\right),\text{where}\begin{array}{cccc}ϵ:& S_n& \to & \left\{\pm 1\right\}\\ & \sigma & \mapsto & det\left(\sigma \right)& .\end{array}$$

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.
2. $$\left|{𝒜}_{\sigma }\right|=\left\{\begin{array}{ll}\frac{\left|{𝒞}_{\sigma }\right|}{2},& \text{if all cycles of} \sigma \text{are of different odd lengths;}\\ \left|{𝒞}_{\sigma }\right|,& \text{otherwise.}\end{array}\right.$$

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 $${\gamma }_{\lambda }=\left(1,2,\dots ,{\lambda }_1\right)\left({\lambda }_1+1,{\lambda }_1+2,\dots ,\right)\left({\lambda }_1+{\lambda }_2+1,\cdots \right)\cdots .$$ 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 $$N=\left\{\left(1234\right),\left(2143\right),\left(3412\right),\left(4321\right)\right\},$$ 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)

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