Notes on Group Theory

David K. Zhang
Last Modified 2022-03-15

Introduction

Group theory is the first subject in a branch of mathematics known as abstract algebra (or modern algebra). Abstract algebra is one of the main branches of modern pure mathematics, and it is a standard requirement in the university mathematics curriculum. Every professional mathematician and theoretically-oriented physical scientist (i.e., physicists and chemists) should know a thing or two about group theory.

The goal of abstract algebra is to study the general properties of algebraic systems and the interrelations between them. That might not mean a lot if you haven’t studied abstract algebra before, so let me contextualize this statement.

In grade school, you learned about a handful of different number systems, including the integers Z\Z, the rational numbers Q\Q, the real numbers R\R, and possibly (if you were an advanced student) the complex numbers C\C. You also learned that there are several algebraic properties that these systems satisfy, 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 ask what other number systems support a notion of addition, negation, multiplication, or some other operation, that satisfies the commutative, associative, distributive, etc. property.

Each of the number systems mentioned above supports the four basic arithmetic operations of addition, subtraction, multiplication, and (with the exception of the integers) division. Of course, subtraction and division are merely the inverses of addition and multiplication, so in each case, there are really only two fundamental arithmetic operations. [TODO: Finish writing introduction.]

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=x1 \cdot x = x \cdot 1 = x for all xGx \in G.
  • Inverse property: xx1=x1x=1x \cdot x^{-1} = x^{-1} \cdot x = 1 for all xGx \in G.

Definition: Subgroup

TODO

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.


Proof:

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}

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

Let GG be a group. An element gGg \in G is central if gg 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){gG:hG, gh=hg} Z(G) \coloneqq \{ g \in G : \forall h \in G,\ gh = hg \}

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).

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).