Alternating group

The alternating group $A_n$ is the Subgroup of the Symmetric group $S_n$ consisting of all even permutations:

$$\boxed{A_n=\ker (\sigma )=\{g\in S_n:\sigma (g)=+1\},}$$

where $\sigma :S_n\to \{\pm 1\}$ is the Signature.

Order and Structure

• $|A_n|=n!/2$ for $n\ge 2$ (half of $S_n$, since signature $\sigma :S_n\to \{\pm 1\}$ is surjective for $n\ge 2$).
• $A_n⊴S_n$ (it’s a Kernel, hence Normal subgroup).
• $S_n/A_n\cong C_2$ (First Isomorphism Theorem with the signature).

Cycle Shapes in $A_n$

A cycle of length $r$ has signature $(-1{)}^{r-1}$. So a permutation with cycle shape $(r_1,r_2,\dots )$ has signature $\prod (-1{)}^{r_i-1}$. Equivalently, the number of even-length cycles must be even for $g\in A_n$.

Cycle shape	$\sigma$	In $A_n$?
$(1^n)$ (identity)	$+1$	✓
$(2)$ (transposition)	$-1$	✗
$(2,2)$ (double transposition)	$+1$	✓
$(3)$ (3-cycle)	$+1$	✓
$(3,2)$	$-1$	✗
$(4)$	$-1$	✗
$(5)$	$+1$	✓

Importance

• Simple for $n\ge 5$: $A_n$ has no nontrivial proper Normal subgroup for $n\ge 5$. This is the algebraic obstruction to solving the general quintic by radicals (Galois theory).
• $A_4$ is not simple - it has the Klein four-group $V_4=\{e,(12)(34),(13)(24),(14)(23)\}$ as a normal subgroup.
• $A_5$ is the smallest nonabelian simple group, with $|A_5|=60$.

Sizes and Examples

• $A_2=\{e\}$ (trivial).
• $A_3=\{e,(123),(132)\}\cong C_3$.
• $A_4$: 12 elements - identity, 8 three-cycles, 3 double transpositions.
• $A_5$: 60 elements - identity, 15 double transpositions, 20 three-cycles, 24 five-cycles.