Permutation Groups $S_n$ and $A_n$

The symmetric group $S_n$ — all permutations of $n$ symbols — is the universal home of finite groups (Cayley's theorem, group-actions.md) and the source of the first non-abelian simple groups $A_n$. Its combinatorics (cycle type, parity) is elementary but decisive: the simplicity of $A_5$ is what makes the general quintic unsolvable (series-solvable.md). This page builds on axioms.md and products.md.

Cycle notation and cycle type

A permutation $\sigma \in S_n$ factors uniquely into disjoint cycles. Disjoint cycles commute, and a $k$-cycle has order $k$, so $\mathrm{ord}(\sigma )$ is the lcm of its cycle lengths. The multiset of cycle lengths is the cycle type, a partition of $n$.

Conjugacy = cycle type. Two permutations are conjugate in $S_n$ iff they have the same cycle type. Hence the conjugacy classes of $S_n$ are indexed by the partitions of $n$, and (via character theory) so are its irreducible representations — the beautiful $S_n$/Young diagram dictionary.

The sign homomorphism and $A_n$

Every permutation is a product of transpositions ($2$-cycles); the parity of the number of transpositions is well-defined, giving the sign homomorphism $$\mathrm{sgn}:S_n\to \{\pm 1\},\mathrm{sgn}(\sigma )=(-1{)}^{\#\text{transpositions}}.$$ A $k$-cycle has sign $(-1{)}^{k-1}$. The kernel is the alternating group $A_n$ of even permutations, a normal subgroup of index $2$ (hence order $n!/2$). Since any transposition splits the sign map, $S_n=A_n⋊{\mathbb{Z}}_2$ (see products.md).

Generation

$S_n$ is generated by small sets:

• all transpositions; or just the adjacent transpositions $(i,i+1)$ (Coxeter generators — see presentations.md);
• a single transposition $(12)$ together with the $n$-cycle $(12\cdots n)$.

$A_n$ is generated by the $3$-cycles, and (for $n\ge 5$) by the products of two disjoint transpositions.

Simplicity of $A_n$

Theorem. $A_n$ is simple for all $n\ge 5$.

$A_5$ (order $60$) is the smallest non-abelian simple group. Proof sketch: the $3$-cycles are all conjugate in $A_n$ for $n\ge 5$ and generate $A_n$; any non-trivial normal subgroup is shown to contain a $3$-cycle, hence all of them, hence everything. The low cases are exceptional: $A_2,A_3$ are abelian (orders $1,3$), and $A_4$ is not simple — it has the normal Klein four-group $V_4=\{e,(12)(34),(13)(24),(14)(23)\}$, the reason $A_4$ has no subgroup of order $6$ (the standard Lagrange-converse counterexample of subgroups-cosets.md).

The simplicity of $A_n$ places it among the finite simple groups (simple-groups.md) and makes $S_n$ unsolvable for $n\ge 5$ — the group-theoretic core of the Abel–Ruffini theorem that the general quintic has no radical solution.

The exceptional outer automorphism of $S_6$

For every $n\ne 6$, all automorphisms of $S_n$ are inner, so $\mathrm{Out}(S_n)=1$. Uniquely, $S_6$ has an outer automorphism, giving $\mathrm{Out}(S_6)\cong {\mathbb{Z}}_2$ — it swaps the class of transpositions with the class of triple transpositions and the two conjugacy classes of $S_5$-subgroups. This exceptional symmetry is tied to the sporadic geometry that later produces the Mathieu groups (see simple-groups.md).

References

• Dummit & Foote, Abstract Algebra, §3.5, §4.6.
• Rotman, An Introduction to the Theory of Groups, Ch. 2–3.
• Fulton & Harris, Representation Theory, Ch. 4 ($S_n$ and Young diagrams).