Notes on Group Theory

David K. Zhang
Last Modified 2024-01-02


Group theory is the first subject in a field of mathematics known as abstract algebra or modern algebra. Abstract algebra is one of the main branches of modern pure mathematics, and every mathematician, scientist, or engineer who wants a firm understanding of theoretical foundations should know a thing or two about group theory.

The goal of abstract algebra is to study the properties of algebraic systems and the interrelations between them. For example, you probably know about a handful of different number systems, including the integers Z\Z, the rational numbers Q\Q, the real numbers R\R, and possibly the complex numbers C\C. You may have also learned about their algebraic properties, such as the commutative property of addition, x+y=y+xx + y = y + x, and the associative property of multiplication, x(yz)=(xy)zx \cdot (y \cdot z) = (x \cdot y) \cdot z.

In abstract algebra, we will go beyond these familiar number systems to study the properties of algebraic systems in general. We will learn how these properties can inform us about the structure of an unknown algebraic system, and we will see how this knowledge can be used to quickly and easily solve problems that would otherwise be difficult or tedious.

To begin our study of abstract algebra, we will focus on one particular type of algebraic system, known as a group. Groups are an ideal starting point because they strike an excellent balance between simplicity and complexity. They are straightforward to define, involving only a single binary operation, and yet they capture most of the key ideas used to study more complicated algebraic structures. In addition, groups are widely used in both pure and applied contexts. Group theory plays a foundational role not only in abstract algebra itself, but also in mathematical analysis, combinatorics, cryptography, physics, chemistry, and materials science.


Definition: Group

A group is an algebraic structure G;1,1,\alg{G; 1, {}^{-1}, \cdot} consisting of:

  • a set GG, called the underlying set;
  • a distinguished element 1G1 \in G, called the identity element;
  • a unary operation 1:GG{}^{-1}: G \to G, written as xx1x \mapsto x^{-1}, called inversion;
  • a binary operation :G×GG\cdot : G \times G \to G, written as (x,y)xy(x, y) \mapsto x \cdot y, called the group operation or group product;

satisfying the following requirements:

  • Associative property: (xy)z=x(yz)(x \cdot y) \cdot z = x \cdot (y \cdot z) for all x,y,zGx, y, z \in G.
  • Identity property: 1x=x1 \cdot x = x for all xGx \in G.
  • Inverse property: x1x=1x^{-1} \cdot x = 1 for all xGx \in G.

The commutative property (xy=yx)(x \cdot y = y \cdot x) is conspicuously absent from the definition of a group.

Identity is the Only Idempotent

Theorem: Let G;1,1,\alg{G; 1, {}^{-1}, \cdot} be a group. The only element xGx \in G having the property that xx=xx \cdot x = x is the identity element.

Proof: First, to see that the identity element 1G1 \in G actually has the claimed property, observe that 11=11 \cdot 1 = 1 follows from the identity property.

Next, to see that no other element of GG has this property, let xGx \in G be given, and suppose xx=xx \cdot x = x. By multiplying both sides by x1x^{-1} on the left, we have x1(xx)=x1xx^{-1} \cdot (x \cdot x) = x^{-1} \cdot x, from which it follows that:

x=1x=(x1x)x=x1(xx)=x1x=1 x = 1 \cdot x = (x^{-1} \cdot x) \cdot x = x^{-1} \cdot (x \cdot x) = x^{-1} \cdot x = 1

Thus, we have proven that xx=xx \cdot x = x implies x=1x = 1.

Right Inverse Property

Theorem: Let G;1,1,\alg{G; 1, {}^{-1}, \cdot} be a group. For all xGx \in G, we have xx1=1x \cdot x^{-1} = 1.

Proof: Let xGx \in G be given. We know, from the definition of a group, that x1x=1x^{-1} \cdot x = 1 and 1x1=x11 \cdot x^{-1} = x^{-1}. It follows that:

xx1=x(1x1)=x((x1x)x1)=x(x1(xx1))=(xx1)(xx1) x \cdot x^{-1} = x \cdot (1 \cdot x^{-1}) = x \cdot ((x^{-1} \cdot x) \cdot x^{-1}) = x \cdot (x^{-1} \cdot (x \cdot x^{-1})) = (x \cdot x^{-1}) \cdot (x \cdot x^{-1})

This shows that the element xx1Gx \cdot x^{-1} \in G remains unchanged when it is multiplied by itself. Using the previous result, this proves that xx1=1x \cdot x^{-1} = 1.

Right Identity Property

Theorem: Let G;1,1,\alg{G; 1, {}^{-1}, \cdot} be a group. For all xGx \in G, we have x1=xx \cdot 1 = x.

Proof: Let xGx \in G be given. We know, from the definition of a group, that x1x=1x^{-1} \cdot x = 1 and 1x=x1 \cdot x = x. It follows that:

x1=x(x1x)=(xx1)x=1x=x x \cdot 1 = x \cdot (x^{-1} \cdot x) = (x \cdot x^{-1}) \cdot x = 1 \cdot x = x

This proves that x1=xx \cdot 1 = x.


Definition: Subgroup, HGH \le G

Let G;1,1,\alg{G; 1, {}^{-1}, \cdot} be a group. A subgroup of G;1,1,\alg{G; 1, {}^{-1}, \cdot} is a subset HGH \subseteq G of its underlying set that satisfies the following requirements:

  • 1H1 \in H.
  • If xHx \in H, then x1Hx^{-1} \in H.
  • If x,yHx, y \in H, then xyHx \cdot y \in H.

We write HGH \le G to denote that HH is a subgroup of G;1,1,\alg{G; 1, {}^{-1}, \cdot}.

In other words, HH is a subgroup of GG if and only if the structure H;1,1H,H\alg{H; 1, {}^{-1}|_H, \cdot|_H} is a group in its own right, where 1H{}^{-1}|_H and H\cdot|_H denote the inverse operation 1{}^{-1} and the group operation \cdot restricted to HH and H×HH \times H, respectively.

This definition introduces an important notational convention. We will often refer to a group G;1,1,\alg{G; 1, {}^{-1}, \cdot} simply by the name of its underlying set GG, omitting explicit mention of the identity element, the inversion operation, and the group operation.

It is also common in the mathematics literature to not use any symbol, such as x×yx \times y or xyx \cdot y, to denote the binary operation in a group. Instead, most mathematicians simply write xyxy, using juxtaposition to indicate application of the group operation. Note that the associative property allows us to write expressions like xyzxyz without needing to specify which product should be evaluated first, since both (xy)z(xy)z and x(yz)x(yz) are guaranteed to produce the same result.

Definition: Commute, Commutes

Let GG be a group. We say that two elements x,yGx, y \in G commute, or we say that xx commutes with yy, if xy=yxxy = yx.

Definition: Central Element, Center, Z(G)Z(G)

Let GG be a group. An element xGx \in G is central if xx commutes with every element of GG. The set of all central elements of GG is called the center of GG, and is denoted by Z(G)Z(G).

Z(G){xG:yG, xy=yx} Z(G) \coloneqq \{ x \in G : \forall y \in G,\ xy = yx \}

Definition: Abelian Group

An abelian group is a group G;1,1,\alg{G; 1, {}^{-1}, \cdot} that satisfies the following additional requirement:

  • Commutative property: xy=yxx \cdot y = y \cdot x for all x,yGx, y \in G.

Definition: Left Coset, Right Coset, G/HG/H, H\GH \backslash G

Let GG be a group, and let HGH \le G be a subgroup. A left coset of HH is a set of the form

gH{gh:hH} gH \coloneqq \{ gh : h \in H \}

for some fixed element gGg \in G. Similarly, a right coset of HH is a set of the form

Hg{hg:hH} Hg \coloneqq \{ hg : h \in H \}

for some fixed element gGg \in G. The collection of all left cosets of HH is denoted by G/HG/H, while the collection of all right cosets of HH is denoted by H\GH \backslash G.

This definition introduces a new notational convention. Whenever we apply a group operation, such as gHgH or H1H^{-1}, to a subset of a group, we mean the set formed by applying that operation to each element of the subset. For example:

gH{gh:hH}H1{h1:hH}g1Hg2{g1hg2:hH} \begin{aligned} gH &\coloneqq \{ gh : h \in H \} \\ H^{-1} &\coloneqq \{ h^{-1} : h \in H \} \\ g_1 H g_2 &\coloneqq \{ g_1 h g_2 : h \in H \} \end{aligned}

Note that we can write g1Hg2g_1 H g_2 without ambiguity, since the associative property guarantees that (g1H)g2=g1(Hg2)(g_1 H) g_2 = g_1 (H g_2).

Definition: Index, G:H\abs{G:H}

Let GG be a group, and let HGH \le G be a subgroup. The index of HH in GG, denoted by G:H\abs{G:H}, is the cardinality of the

Cosets Partition a Group

Theorem: Let GG be a group, and let HGH \le G be a subgroup.


Suppose two cosets xHxH and yHyH of HH intersect, i.e., there exists an element zxHyHz \in xH \cap yH

Definition: Normal Subgroup

Let GG be a group. We say that a subgroup HGH \le G is normal if for all gGg \in G and hHh \in H, we have g1hgHg^{-1}hg \in H.

left conjugation — this is a left action

gxgxg1 {}^g x \coloneqq g x g^{-1} h(gx)=h(gxg1)=h(gxg1)h1=(hg)x(hg)1=hgx {}^h ({}^g x) = {}^h (g x g^{-1}) = h (g x g^{-1}) h^{-1} = (hg) x (hg)^{-1} = {}^{hg} x

right conjugation — this is a right action

xgg1xg x^g \coloneqq g^{-1} x g (xg)h=(g1xg)h=h1(g1xg)h=(gh)1x(gh)=xgh (x^g)^h = (g^{-1} x g)^h = h^{-1} (g^{-1} x g) h = (gh)^{-1} x (gh) = x^{gh}

Center is a Subgroup

Theorem: Let GG be a group. Its center Z(G)Z(G) is a subgroup of GG.

Proof: We must show that Z(G)Z(G) contains the identity element 11 and is closed under taking inverses and products.

  • Clearly, 1Z(G)1 \in Z(G), since the identity element 11 commutes with everything.
  • Let aZ(G)a \in Z(G) and gGg \in G. By definition, we have ag=gaag = ga. By multiplying on the left and right by a1a^{-1}, we obtain ga1=a1gga^{-1} = a^{-1}g. Hence, a1a^{-1} commutes with gg. Since gGg \in G was arbitrary, this proves that a1Z(G)a^{-1} \in Z(G).
  • Let a,bZ(G)a, b \in Z(G) and gGg \in G. Because aa and bb are central, we can write abg=agb=gababg = agb = gab. Hence, abab commutes with gg. Again, because gGg \in G was arbitrary, this proves that abZ(G)ab \in Z(G).

Center is a Normal Subgroup

Theorem: Let GG be a group. Its center Z(G)Z(G) is a normal subgroup of GG.

Proof: Let gGg \in G and zZ(G)z \in Z(G). Because zz is central, we have g1zg=zg1g=zZ(G)g^{-1}zg = zg^{-1}g = z \in Z(G).