1 The Group Axioms
A group is a pair (G,⋅) consisting of a set G and a binary operation ⋅:G×G→G satisfying the following three axioms:
Associativity. For all a,b,c∈G, (a⋅b)⋅c=a⋅(b⋅c).
Identity element. There exists an element e∈G such that e⋅a=a⋅e=a for all a∈G.
Inverses. For every a∈G, there exists an element b∈G such that a⋅b=b⋅a=e.
Throughout, we write the group operation by juxtaposition (ab instead of a⋅b), denote the identity element by e, and denote the inverse of a by a−1.
A group (G,⋅) is called abelian (or commutative) if ab=ba for all a,b∈G.
2 Fundamental Examples
Let us ground the abstract definition with some concrete examples.
(Z,+) is a group. Associativity is clear, the identity element is 0, and the inverse of a is −a. Since a+b=b+a, it is abelian.
GLn(R)={A∈Mn(R)∣detA=0} is a group under matrix multiplication. The identity element is the identity matrix In, and the inverse is the matrix inverse A−1. For n≥2, there exist matrices A,B with AB=BA, so this group is non-abelian.
Let Sn denote the set of all bijections from {1,2,…,n} to itself. Under composition of maps, Sn forms a group called the symmetric group onnletters. Its order is ∣Sn∣=n!, and it is non-abelian for n≥3.
For a positive integer n, define Z/nZ={0ˉ,1ˉ,…,n−1} with the operation aˉ+bˉ=a+b. This is an abelian group.
The set {e} with e⋅e=e forms a group, called the trivial group.
3 Uniqueness of the Identity and Inverses
The group axioms only assert the existence of an identity element and of inverses. However, they are in fact unique.
The identity element of a group G is unique.
Suppose e and e′ are both identity elements. Since e is an identity, e⋅e′=e′. Since e′ is an identity, e⋅e′=e. Therefore e=e′. □
For each element a of a group G, the inverse a−1 is unique.
Suppose b and c are both inverses of a, so that ab=ba=e and ac=ca=e. Then b=be=b(ac)=(ba)c=ec=c.
□
For any element a of a group G, (a−1)−1=a.
The inverse of a−1 is the element x satisfying a−1x=xa−1=e. Since a−1a=aa−1=e, we have x=a. □
For any elements a,b of a group G, (ab)−1=b−1a−1.
We verify directly: (ab)(b−1a−1)=a(bb−1)a−1=aea−1=aa−1=e. Similarly, (b−1a−1)(ab)=e. □
4 Cancellation Laws
In a group G, the following hold:
Left cancellation:ab=ac⇒b=c.
Right cancellation:ba=ca⇒b=c.
(1) If ab=ac, multiply both sides on the left by a−1: a−1(ab)=a−1(ac)⟹(a−1a)b=(a−1a)c⟹eb=ec⟹b=c.
(2) Similarly, multiply on the right by a−1. □
In a group G, for any a,b∈G, the equations ax=b and ya=b each have a unique solution.
The equation ax=b has the solution x=a−1b. Indeed, a(a−1b)=(aa−1)b=eb=b. Uniqueness follows from the cancellation law. Similarly, y=ba−1 is the unique solution to ya=b. □
5 Powers and Order of an Element
For an element a of a group G and an integer n, we define the poweran as follows:
For an element a of a group G and integers m,n, the following hold:
am+n=aman.
(am)n=amn.
(1) When m,n≥0, this follows by induction on n. When m<0 and n≥0 with ∣m∣≤n, we have aman=(a−1)∣m∣an=an−∣m∣=am+n. The remaining cases are handled by similar case analysis. (2) Follows by the same method. □
For an element a of a group G, if there exists a smallest positive integer n such that an=e, then n is called the order of a, and we write ord(a)=n or ∣a∣=n. If no such positive integer exists, we say a has infinite order and write ord(a)=∞.
In Z/6Z, we have ord(1ˉ)=6, ord(2ˉ)=3, ord(3ˉ)=2, and ord(0ˉ)=1.
In (Z,+), every nonzero element has infinite order.
Let ord(a)=n<∞. Then ak=e if and only if n∣k.
(⇐): If k=nq, then ak=(an)q=eq=e.(⇒): Write k=nq+r with 0≤r<n. From ak=e we get ar=ak−nq=ak(an)−q=e⋅e=e. Since r<n and n is the smallest positive integer with an=e, we must have r=0, i.e. n∣k. □
6 Subgroups
A nonempty subset H of a group G is called a subgroup of G if H is itself a group under the operation of G. We write H≤G. If H={e} and H=G, we say H is a proper subgroup and write H<G.
Checking the subgroup axioms directly from the definition is often cumbersome. The following criterion provides a convenient shortcut.
A nonempty subset H of a group G is a subgroup if and only if ab−1∈H for all a,b∈H.
(⇒): If H is a subgroup, then b∈H implies b−1∈H, and then a,b−1∈H implies ab−1∈H.(⇐): Since H=∅, pick any a∈H. Taking b=a gives aa−1=e∈H. Taking a=e gives eb−1=b−1∈H for any b∈H. For a,b∈H, since b−1∈H, we have a(b−1)−1=ab∈H (closure). Associativity is inherited from G. □
A nonempty finite subset H of a group G is a subgroup if and only if H is closed under the group operation (i.e. a,b∈H⇒ab∈H).
(⇒) is clear. (⇐): Let a∈H. The elements a,a2,a3,… all lie in H, and since H is finite, we must have ai=aj for some i<j. By cancellation, aj−i=e, so e∈H. If j−i≥2, then aj−i−1=a−1∈H. If j−i=1, then a=e, so a−1=e∈H. □
For any group G, the subsets {e} and G are subgroups (trivial subgroups).
The subgroups of (Z,+) are precisely the sets nZ={nk∣k∈Z} for n=0,1,2,… (proved below).
SLn(R)={A∈GLn(R)∣detA=1} is a subgroup of GLn(R) (the special linear group).
The set An of all even permutations is a subgroup of Sn (the alternating group), with ∣An∣=n!/2.
If {Hi}i∈I is a family of subgroups of G, then ⋂i∈IHi is also a subgroup of G.
Since e∈Hi for all i, we have ⋂Hi=∅. If a,b∈⋂Hi, then a,b∈Hi for each i, and since each Hi is a subgroup, ab−1∈Hi. Therefore ab−1∈⋂Hi. □
7 Generators and Cyclic Groups
For a subset S of a group G, the subgroup generated byS is defined as the intersection of all subgroups of G containing S: ⟨S⟩=S⊆H≤G⋂H
⟨S⟩ equals the set of all finite products of elements of S and their inverses: ⟨S⟩={a1ε1a2ε2⋯akεk∣k≥0,ai∈S,εi=±1}
where the empty product (when k=0) is defined to be e.
Let T denote the right-hand side. We must verify three things: that T is a subgroup, that S⊆T, and that T is contained in every subgroup that contains S.T is closed under the group operation (concatenate two products). We have e∈T (take k=0). The inverse of a1ε1⋯akεk is ak−εk⋯a1−ε1∈T. Thus T is a subgroup.For a∈S, taking k=1 and ε1=1 gives a∈T, so S⊆T.If S⊆H≤G, then H is closed under products and inverses, so T⊆H. Therefore T=⟨S⟩. □
A group G is cyclic if G=⟨a⟩ for some element a∈G. The element a is called a generator of G.
The cyclic group ⟨a⟩ is precisely the set {an∣n∈Z} of all powers of a.
If ord(a)=∞, then ⟨a⟩≅Z (as additive groups).
If ord(a)=n<∞, then ⟨a⟩={e,a,a2,…,an−1}, ∣⟨a⟩∣=n, and ⟨a⟩≅Z/nZ.
(1) Define φ:Z→⟨a⟩ by φ(k)=ak. Then φ(m+n)=am+n=aman=φ(m)φ(n), so φ is a homomorphism. Since ord(a)=∞, if am=an then am−n=e, which forces m=n. So φ is injective. Surjectivity is immediate from the definition. Hence φ is an isomorphism.(2) We first show e,a,a2,…,an−1 are all distinct. If ai=aj with 0≤i<j≤n−1, then aj−i=e with 0<j−i<n, contradicting the minimality of n. So ∣⟨a⟩∣=n. The isomorphism is given by φ:Z/nZ→⟨a⟩, φ(kˉ)=ak; well-definedness follows from the theorem relating order and powers. □
Every subgroup of a cyclic group is cyclic.
Let G=⟨a⟩ and let H≤G. If H={e}, then H=⟨e⟩ is cyclic. Suppose H={e}. Let d be the smallest positive integer such that ad∈H. (Such a d exists because H contains some ak with k=0, and ak∈H implies a−k∈H, so we can always find a positive power.)If am∈H, write m=dq+r with 0≤r<d. Then ar=am(ad)−q∈H. By the minimality of d, we must have r=0, i.e. d∣m. Therefore H=⟨ad⟩. □
For a cyclic group G=⟨a⟩ of order n, the element ak is a generator of G if and only if gcd(k,n)=1. Consequently, G has exactly φ(n) generators, where φ is Euler's totient function.
We show that ord(ak)=n/gcd(k,n). Let d=gcd(k,n). We have (ak)n/d=akn/d=(an)k/d=e, so ord(ak)∣n/d. Conversely, if (ak)m=e, then n∣km, i.e. (n/d)∣(k/d)m. Since gcd(n/d,k/d)=1, it follows that (n/d)∣m. Therefore ord(ak)=n/d. Now ord(ak)=n if and only if d=1, i.e. gcd(k,n)=1. □
8 Order of a Group
The order of a group G, denoted ∣G∣, is the cardinality of the underlying set. If ∣G∣<∞, we call G a finite group; otherwise, G is an infinite group.
∣Sn∣=n!, ∣Z/nZ∣=n, ∣Dn∣=2n (the dihedral group of a regular n-gon), and ∣(Z,+)∣=∞.
For an element a of a finite group G, ord(a) divides ∣G∣. In particular, a∣G∣=e.
The cyclic subgroup ⟨a⟩ is a subgroup of G with ∣⟨a⟩∣=ord(a). By Lagrange's theorem (proved in the next chapter), ∣⟨a⟩∣ divides ∣G∣. Writing ∣G∣=ord(a)⋅m, we get a∣G∣=(aord(a))m=em=e. □
9 Fundamentals of Group Homomorphisms
We will study homomorphisms in depth later, but it is convenient to state the definition and basic properties now.
A map φ:G→G′ between groups is called a group homomorphism if φ(ab)=φ(a)φ(b)for all a,b∈G.
Let φ:G→G′ be a group homomorphism. Then:
φ(eG)=eG′.
φ(a−1)=φ(a)−1.
φ(an)=φ(a)n for all n∈Z.
(1) We have φ(eG)=φ(eG⋅eG)=φ(eG)φ(eG). Multiplying both sides by φ(eG)−1 yields eG′=φ(eG).(2) φ(a)φ(a−1)=φ(aa−1)=φ(eG)=eG′, so φ(a−1)=φ(a)−1.(3) For n≥0, this follows by induction. For n<0, use part (2). □
For a group homomorphism φ:G→G′:
One can verify that kerφ≤G and Imφ≤G′.
A group homomorphism φ:G→G′ is injective if and only if kerφ={eG}.
(⇒): If φ(a)=eG′=φ(eG), then injectivity gives a=eG.(⇐): If φ(a)=φ(b), then φ(ab−1)=φ(a)φ(b)−1=eG′, so ab−1∈kerφ={eG}, whence a=b. □
10 Summary and Next Steps
In this chapter we have systematically developed the foundations of group theory:
The group axioms and fundamental examples.
Uniqueness of the identity and inverses, cancellation laws, and solubility of equations.
Laws of exponents and the order of an element.
The definition of subgroups and criteria for recognizing them.
The structure of cyclic groups and their subgroups.
Group homomorphisms: definition, kernel, image, and the injectivity criterion.
In the next chapter, we consider how a subgroup "partitions" a group. This leads to the concept of cosets and to Lagrange's theorem— the most fundamental theorem in the theory of finite groups.