Symmetric Group in Group Theory: A Complete Guide

The signature (or sign) of a permutation is +1 if the permutation is even, and -1 if the permutation is odd. It is a homomorphism from the symmetric group to the multiplicative group {+1, -1}. The signature is equal to (-1)^k, where k is the number of inversions in the permutation.

Permutations decompose into transpositions (swaps of two elements), classified as even or odd based on the number of swaps. The signature (+1 for even, -1 for odd) is a group homomorphism: the signature of a composition equals the product of individual signatures. This algebraic structure underlies alternating forms, as each transposition introduces a sign change. Understanding permutation parity provides the mathematical machinery needed to handle higher-dimensional determinants systematically.

The alternating group A_n is the subgroup of the symmetric group S_n containing only even permutations, which are permutations that can be expressed as a product of an even number of transpositions; a transposition (swapping two elements) is always an odd permutation, and the parity of a cycle of length k is even if k is odd and odd if k is even, with the composition rules being: even × even = even, even × odd = odd, odd × even = odd, and odd × odd = even.

The parity of a permutation is determined by the number of inversions (pairs of indices where a larger element precedes a smaller one) in its lower row: if the number of inversions is even, the permutation is even; if odd, it is odd. The composition of permutations follows the same parity rules as adding even and odd numbers: even + even = even, odd + odd = even, and even + odd = odd. This parity function is a group homomorphism from the symmetric group to the group of two elements (even and odd). The set of all even permutations forms a subgroup of the symmetric group, called the alternating group.

This section explores permutations and their classification. Any permutation can be decomposed into transpositions (swaps). If a permutation requires an even number of transpositions, it is called even; if odd, it is called odd. The alternating group A₅ consists of all even permutations of five elements and forms a group because composing two even permutations yields another even permutation. A three-dimensional representation of A₅ maps elements to 3×3 complex matrices. A 3-cycle maps to a rotation matrix involving the golden ratio φ, while a double transposition maps to another rotation matrix. Both types of matrices are orthogonal with determinant 1, meaning they only rotate space without distorting it.

The number of conjugacy classes in the symmetric group Sₙ is equal to the number of integer partitions of n, where a partition of n is a way of writing n as a sum of positive integers (order does not matter). For example, the number of conjugacy classes in S₆ is 11 because there are 11 partitions of 6.

The number of conjugacy classes in Sₙ equals the number of partitions of n. Each partition of n corresponds to a unique cycle type, and each cycle type corresponds to a unique conjugacy class. This establishes a fundamental correspondence between integer partitions and conjugacy classes in symmetric groups.

In symmetric groups S_n, conjugacy classes are in one-to-one correspondence with integer partitions of n. Each partition corresponds to cycle types: for example, in S₄, the partitions 4, 3+1, 2+2, 2+1+1, and 1+1+1+1 correspond to 4-cycles, 3-cycles, double transpositions, transpositions, and identity respectively. The number of elements in each conjugacy class can be calculated combinatorially based on the cycle type.

In symmetric groups S_n, elements are classified by cycle structure, described by lengths of disjoint cycles summing to n. The size of a conjugacy class equals n! divided by the product of (k_i × m_i!), where k_i are cycle lengths and m_i are their multiplicities. The centralizer order is n! divided by the conjugacy class size, following |C_G(g)| = |G| / |Cl(g)|. The number of conjugacy classes in S_n equals the number of partitions of n. For S_4, classes correspond to partitions: identity (1), transpositions (6), double transpositions (3), 3-cycles (8), and 4-cycles (6). For S_5, class sizes are calculated using the formula, yielding 30 for structure 2+2+1+1 and 20 for structure 3+2+1. The number of permutations of length r in S_n is n! / (n-r)!.

In the symmetric group S_n, two permutations are conjugate if and only if they correspond to the same partition of n. A partition is a sequence of strictly positive integers in non-increasing order summing to n. For example, in S_8, the partition 3+2+2+1 corresponds to a specific conjugacy class. The multiplicity notation m_k counts how many parts of size k appear in the partition, with Σ(m_k × k) = n. The conjugacy classes partition S_n, and their cardinalities sum to n!.

For group actions: orbit of m is {g·m | g∈G}, stabilizer is {g∈G | g·m=m}. The orbit-stabilizer theorem states |Stab(m)| × |Orb(m)| = |G|. This shows larger orbits correspond to smaller stabilizers. For a cube's vertex, orbit has 8 vertices, stabilizer has 3 rotations (0°, 120°, 240°), giving 24 total rotational symmetries.

The stabilizer of an element a is the set of all g in G such that φ(g, a) = a, denoted G_a. The kernel equals the intersection of stabilizers of all elements in A. The orbit of an element a is the set of all elements reachable from a: Orb(a) = {φ(g, a) | g in G}. The Orbit-Stabilizer Theorem states that |Orb(a)| = [G : G_a], where [G : G_a] is the index of G_a in G. Elements in the same orbit have the same orbit.

A group G acts on a set X if there is a mapping G × X → X satisfying (g1g2)x = g1(g2x) and ex = x. Key concepts include orbits (Gx = {gx | g ∈ G}) and stabilizers (Stab(x) = {g ∈ G | gx = x}). Examples include: conjugation action on G (orbits are conjugacy classes, stabilizers are centralizers), S_n acting on {1,...,n} (transitive, stabilizer is S_{n-1}), isometry group of plane (orbits are origin and circles), and action on left cosets of H (transitive, stabilizer is xHx⁻¹). The Orbit-Stabilizer Theorem establishes a natural bijection between the orbit of x and the coset space G/Stab(x), given by gStab(x) ↦ gx. This implies |Orb(x)| = [G : Stab(x)]. For conjugation action, this gives |Cl(x)| = [G : C_G(x)].

A group action of a group G on a set X is a map G × X → X satisfying e·x = x and a·(b·x) = (ab)·x for all a, b ∈ G and x ∈ X, which is equivalent to a group homomorphism from G to the symmetric group SX; key concepts include invariant subsets, transitive actions (where the only invariant subsets are X and ∅), orbits (the set of all elements reachable from a given element via group action), and stabilizers (subgroups fixing a particular element), with the fundamental orbit-stabilizer theorem stating that for any element m in X, the size of its orbit equals the index of its stabilizer in G, i.e., |Orb(m)| = |G| / |Stab(m)|.

A group G acts on a set X via a map satisfying identity and compatibility axioms, inducing permutations of X. The orbit of x is the set of elements x can be mapped to; the stabilizer is the subgroup fixing x. The Orbit-Stabilizer Theorem states |Orb(x)|·|Stab(x)| = |G|. The stabilizer is a subgroup because it contains identity, is closed under inverses, and under multiplication. The proof constructs a bijection between left cosets of Stab(x) and the orbit. This theorem is fundamental for understanding how groups transform sets and counting orbits.

The symmetric group S₅ contains 120 permutations of five elements. Its subgroup A₅ (even permutations) contains 60 elements and is generated by all 5-cycles. A critical observation is that any 5-cycle must act trivially on any expression taking only 4 values, because applying a 5-cycle five times returns to identity while the expression would have cycled through 5 values. Since A₅ has index 2 in S₅, any expression invariant under A₅ must be invariant under all of S₅, making it impossible to construct a resolvent quartic for quintics as was possible for lower degrees.

While quadratic, cubic, and quartic equations can be solved using formulas built from arithmetic operations and nth roots, the general quintic equation (degree 5) cannot be solved by such formulas. This mystery puzzled mathematicians for over 350 years until Evariste Galois solved it in 1830. Galois revolutionized algebra by realizing that solving equations is fundamentally about studying symmetries rather than grinding through formulas. He showed that if the Galois group (the group of symmetries of the roots) forms a cyclic tower, then the roots can be expressed using radicals. However, if the symmetries are more complex and not cyclic, no combination of the allowed operations can express all the roots. For the general quintic over rationals, the Galois group is S5 (the full set of permutations of five objects), which contains A5—a simple group with no smaller building blocks. This structural rigidity means no formula in radicals can capture the quintic's roots. The solutions can be expressed using elliptic modular functions, which encode the symmetries of A5.

The symmetric group S5 can be generated by a single transposition and a single 5-cycle. By conjugating the transposition by powers of the 5-cycle, we obtain all adjacent transpositions. Then, by conjugating these transpositions by each other, we can obtain any transposition involving any pair of elements. Since all permutations can be written as products of transpositions, this generates the entire symmetric group S5. This completes the proof that the Galois group is indeed S5, and since S5 is not solvable, the polynomial is not solvable by radicals.

Not all Galois groups are solvable. The symmetric group S₅ (all permutations of 5 roots) cannot be built as a series of cyclic extensions. This is why general quintic equations (degree 5) and higher are not solvable by radicals—they lack formulas expressible in terms of radicals. The breakthrough of Galois theory was proving that an equation is solvable by radicals if and only if its Galois group is solvable.