Last updates: 8 December 2010

**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; $${A}_{n}=ker\left(\u03f5\right),\phantom{\rule{3em}{0ex}}\text{where}\phantom{\rule{3em}{0ex}}\begin{array}{cccc}\u03f5:& {S}_{n}& \to & \left\{\pm 1\right\}\\ & \sigma & \mapsto & det\left(\sigma \right)& .\end{array}$$

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

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

- The conjugacy class ${\mathcal{C}}_{\lambda}$ is contained in ${A}_{n}$ if an even number of the ${\lambda}_{i}$ are even numbers.
- If the parts ${\lambda}_{i}$ of $\lambda $ are all odd and are all distinct then ${\mathcal{C}}_{\lambda}$ is a union of two conjugacy classes of ${A}_{n}$ and these two conjugacy classes have the same size.
- Otherwise ${\mathcal{C}}_{\lambda}$ is also a conjugacy class of ${A}_{n}$.

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

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

Let $\sigma \in {A}_{n}$, and let
$\lambda =\left({\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{k}\right)$
be the cycle type of $\sigma $. Let

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

- If $n\ne 4$ then ${A}_{n}$ is simple.
- 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.

Suppose $N$ is a normal subgroup of ${A}_{n}$

[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)