Ideas in finite group theory

Ideas in finite group theory

Martino Garonzi

Abstract

In this note I present some of the main ideas of finite group theory, starting with examples of non-abelian groups (groups of matrices and groups of permutations), going to Galois theory, i.e. the way polynomials and groups interact, and finally simple groups, solvable groups and their role in understanding when the roots of a polynomial can be expressed by starting with the coefficients and performing sums, differences, products, divisions and root extractions.

When dealing with operations there are two possible notations, the additive notation and the multiplicative notation. In the additive notation the operation between two group elements a, b is denoted a+b and the identity element is denoted 0. In the multiplicative notation the operation between two group elements a, b is denoted a·b or simply ab and the identity element is denoted 1. Usually the additive notation is reserved for the abelian case. I will mostly use the multiplicative notation. I will assume the reader to be familiar with the basic properties of groups and fields. The notation H ≤ G means that H is a subgroup of G, and the notation H ⫳ G means that H is a normal subgroup of G.

The given bibliography provides good reference books for the theory of (finite) groups.

I will start by recalling the isomorphism theorem.

Theorem 1 (Isomorphism theorem)

Let ϕ : G → H be a group homomorphism and let N := ker(ϕ) be the kernel of ϕ, i.e. the set of elements g ∈ G such that ϕ(g) = 1. Then G/N ≅ ϕ(G) via the canonical isomorphism gN ↦ ϕ(g).

1. Some examples of groups

Usually abelian groups (commutative groups), i.e. groups in which any two elements a, b verify ab = ba (“commute”), are familiar to every mathematician. Let us start with some examples of non-abelian groups.

1.1. Matrices

Invertible matrices form a group. Let F be any field (for example Q, R, C, Z/pZ). I denote by GL(n, F) the set of n × n invertible matrices with entries in the field F. The usual row-column multiplication gives GL(n, F) the structure of a group, which is non-abelian if n ≥ 2. It is usually called the “General Linear Group”. It is non-abelian because, as it is well-known, the row-column multiplication is not a commutative operation.

The set F^∗ = F − {0} is a group with respect to multiplication, and it is abelian, isomorphic to GL(1, F). Taking the determinant provides a (surjective!) group homomorphism GL(n, F) → F^∗, A ↦ det(A) (indeed det(AB) = det(A) det(B) by Binet’s theorem) whose kernel, SL(n, F) := {A ∈ GL(n, F) : det(A) = 1}, is called the “Special Linear Group”. It is a normal subgroup of GL(n, F) (being the kernel of a homomorphism) and using the isomorphism theorem we see that the quotient GL(n, F)/SL(n, F) is isomorphic to F^∗.

1.2. Permutations

I will denote by Sym(X) the set of bijections X → X (also called the permutations of X). The operation of usual composition of functions gives Sym(X) the structure of group. It is called the “Symmetric Group” of X.

It is an easy exercise to show that if X, Y are equipotent sets then Sym(X) and Sym(Y) are isomorphic groups. If X = {1, . . . , n} I shall denote Sym(X) by Sym(n) or S_n. It is called the symmetric group of degree n. This group is non-abelian if and only if n ≥ 3. An element of Sym(n) is called permutation of {1, . . . , n}. The order of Sym(n) (its size as a set) is n! = 1 · 2 · · · n. I will use the standard cycle notation, which is best explained by means of examples:

(123)(4567) : 1 → 2, 2 → 3, 3 → 1, 4 → 5, 5 → 6, 6 → 7, 7 → 4.

(123 · · · k) : 1 → 2, 2 → 3, 3 → · · · → k → 1, a k-cycle.

Composition goes as follows:

(12)(234)(13) = (234), (143)(1352)(4312) = (13)(45).

Note that disjoint cycles always commute. The following calculation shows that Sym(n) is non-abelian for n ≥ 3:

(12)(123) = (13), (123)(12) = (23).

Remark 1. Every permutation can be written uniquely (up to reordering) as product of disjoint cycles.

2-cycles are also called “transpositions”. A permutation is called “even” (or “of sign 1”) if it can be written as the product of an even number of transpositions, and “odd” (or “of sign −1”) otherwise. For example (12)(25)(13)(35) is even, (13)(26)(43) is odd. The identity of Sym(n) (the identity function {1, . . . , n} → {1, . . . , n}) is considered to be the product of zero transpositions, hence an even permutation.

Remark 2. A product of disjoint cycles is an even permutation if and only if the number of cycles of even length is even.

For example (123)(4567), (12)(3456)(78) are odd, (123)(45)(67), (123)(4567)(89) are even.

Definition 1

The “cycle structure” of a permutation is the increasing sequence of the cycle lengths in the representation as a product of disjoint cycles. Cycles of length 1 are usually omitted.

So for example (123)(4567), (12)(3456)(78), (123)(45)(67), (123)(4567)(89) have cycle structure respectively (3, 4), (2, 2, 4), (2, 2, 3), (2, 3, 4). Remark 2 implies that the cycle structure of a permutation determines its sign. Hence all elements of cycle structure (3, 4), (2, 2, 4) are odd, and all elements of cycle structure (2, 2, 3), (2, 3, 4) are even.

Let us denote by C₂ the set {1, −1} with the operation given by multiplication: 1 · 1 = (−1) · (−1) = 1 and 1 · (−1) = (−1) · 1 = −1. Then C₂ is a commutative group, it is a cyclic group (a group generated by one element) of order 2 (generated by −1), and it is isomorphic to Sym(2) = {1, (12)}. Consider the map

sgn : Sym(n) → {−1, 1} = C₂, σ ↦ sgn(σ)

which sends any permutation to its sign (1 if it is even, −1 if it is odd). Then sgn is a (surjective!) group homomorphism whose kernel, Alt(n) = A_n := {σ ∈ Sym(n) : sgn(σ) = 1}, is called the “Alternating Group” of degree n. It is a normal subgroup of Sym(n) (being the kernel of a homomorphism) and using the isomorphism theorem we see that the quotient Sym(n)/Alt(n) is isomorphic to C₂, in particular |Alt(n)| = n!/2.

For example Alt(4) = {(12)(34), (13)(24), (14)(23), (123), (132), (124), (142), (234), (243), (134), (143)}.

2. Galois theory

2.1. The Galois group of a polynomial

Let us talk about the reason why groups were invented.

To each polynomial f ∈ Q[X] without multiple roots can be attached a finite group G_f, called the Galois group of the polynomial (named after Evariste Galois, 1811–1832). It is the group defined as follows: if a₁, . . . , a_n denote the distinct roots of f(X) then

G_f = Aut(Q(a₁, . . . , a_n))

where Q(a₁, . . . , a_n) denotes the field generated by a₁, . . . , a_n, i.e. the intersection of the subfields of C containing a₁, . . . , a_n. That is, G_f is the group of field isomorphisms Q(a₁, . . . , a_n) → Q(a₁, . . . , a_n), that is, the group of such maps which are bijective and respect identity elements, sums, and products. It goes without saying that the operation in G_f is again the usual composition of functions.