# Elementary group theory

In mathematics, a group (G,*) is usually defined as:

G is a set and * is an associative binary operation on G, obeying the following rules (or axioms):

A1. ( Closure) If a and b are in G, then a*b is in G
A2. ( Associativity) If a, b, and c are in G, then (a*b)*c=a*(b*c).
A3. ( Identity) G contains an element, often denoted e, such that for all a in G, a*e=e*a=a. We call this element the identity of (G,*). (We will show e is unique later.)
A4. ( Inverses) If a is in G, then there exists an element b in G such that a*b=b*a=e. We call b the inverse of a. (We will show b is unique later.)

Closure and associativity are part of the definition of "associative binary operation", and are sometimes omitted, particularly closure.

Notes:

• The * is not necessarily multiplication. Addition works just as well, as do many less standard operations.
• When * is a standard operation, we use the standard symbol instead (for example, + for addition).
• When * is addition or any commutative operation (except multiplication), the identity is usually denoted by 0 and the inverse of a by -a. The operation is always denoted by something other than *, often +, to avoid confusion with multiplication.
• When * is multiplication or any non-commutative operation, the identity is usually denoted by 1 and the inverse of a by a -1. The operation is often omitted, a*b is often written ab.
• (G,*) is usually pronounced "the group G under *". When affirming that it is a group (for example, in a theorem), we say that "G is a group under *".
• The group (G,*) is often referred to as "the group G" or simply "G"; but the operation "*" is fundamental to the description of the group.

## Examples

### (R,+) is a group

The real numbers (R) are a group under addition (+).

Closure: Clear; adding any two numbers gives another number.
Associativity: Clear; for any a, b, c in R, (a+b)+c=a+(b+c).
Identity: 0. For any a in R, a+0=a. (Hence the denotation 0 for identity)
Inverses: For any a in R, -a+a=0. (Hence the denotation -a for inverse)

### (R,*) is not a group

The real numbers (R) are NOT a group under multiplication (*).

Identity: 1.
Inverses: 0*a=0 for all a in R, so 0 has no inverse.

### (R#,*) is a group

The real numbers without 0 (R#) are a group under multiplication (*).

Closure: Clear; multiplying any two numbers gives another number.
Associativity: Clear; for any a, b, c in R, (a*b)*c=a*(b*c).
Identity: 1. For any a in R, a*1=a. (Hence the denotation 1 for identity)
Inverses: For any a in R, a -1*a=1. (Hence the denotation a -1 for inverse)

## Basic theorems

### Inverse relations are commutative

Theorem 1.1: For all a in G, a -1*a = e.

• By expanding a -1*a, we get
• a -1*a = a -1*a*e (by A3')
• a -1*a*e = a -1*a*(a -1*(a -1) -1) (by A4', a -1 has an inverse denoted (a -1) -1)
• a -1*a*(a -1*(a -1) -1) = a -1*(a*a -1)*(a -1) -1 = a -1*e*(a -1) -1 (by associativity and A4')
• a -1*e*(a -1) -1 = a -1*(a -1) -1 = e (by A3' and A4')
• Therefore, a -1*a = e

### Identity relations are commutative

Theorem 1.2: For all a in G, e*a = a.

• Expanding e*a,
• e*a = (a*a -1)*a (by A4)
• (a*a -1)*a = a*(a -1*a) = a*e (by associativity and the previous theorem)
• a*e = a (by A3)
• Therefore e*a = a

### Latin square property

Theorem 1.3: For all a,b in G, there exists a unique x in G such that a*x = b.

• Certainly, at least one such x exists, for if we let x = a -1*b, then x is in G (by A1, closure); and then
• a*x = a*(a -1*b) (substituting for x)
• a*(a -1*b) = (a*a -1)*b (associativity A2).
• (a*a -1)*b= e*b = b. (identity A3).
• Thus an x always exists satisfying a*x = b.
• To show that this is unique, if a*x=b, then
• x = e*x
• e*x = (a -1*a)*x
• (a -1*a)*x = a -1*(a*x)
• a -1*(a*x) = a -1*b
• Thus, x = a -1*b

Similarly, for all a,b in G, there exists a unique y in G such that y*a = b.

### The identity is unique

Theorem 1.4: The identity element of a group (G,*) is unique.

• a*e = a (by A3)
• Apply theorem 1.3, with b = a.

Alternative proof: Suppose that G has two identity elements, e and f say. Then e*f = e, by A3', but also e*f = f, by Theorem 1.2. Hence e = f.

As a result, we can speak of the identity element of (G,*) rather than an identity element. Where different groups are being discussed and compared, often eG will be used to identify the identity in (G,*).

### Inverses are unique

Theorem 1.5: The inverse of each element in (G,*) is unique; equivalently, for all a in G, a*x = e if and only if x=a -1.

• If x=a -1, then a*x = e by A4.
• Apply theorem 1.3, with b = e.

Alternative proof: Suppose that an element g of G has two inverses, h and k say. Then h = h*e = h*(g*k) = (h*g)*k = e*k = k (equalities justified by A3'; A4'; A2; Theorem 1.1; and Theorem 1.2, respectively).

As a result, we can speak of the inverse of an element x, rather than an inverse.

### Inverting twice gets you back where you started

Theorem 1.6: For all a belonging to a group (G,*), (a -1) -1=a.

• a -1*a = e.
• Therefore the conclusion follows from theorem 1.4.

### The inverse of ab

Theorem 1.7: For all a,b belonging to a group (G,*), (a*b) -1=b -1*a -1.

• (a*b)*(b -1*a -1) = a*(b*b -1)*a -1 = a*e*a -1 = a*a -1 = e
• Therefore the conclusion follows from theorem 1.4.

### Cancellation

Theorem 1.8: For all a,x,y, belonging to a group (G,*), if a*x=a*y, then x=y; and if x*a=y*a, then x=y.

• If a*x = a*y then:
• a -1*(a*x) = a -1*(a*y)
• (a -1*a)*x = (a -1*a)*y
• e*x = e*y
• x = y
• If x*a = y*a then
• (x*a)*a -1 = (y*a)*a -1
• x*(a*a -1) = y*(a*a -1)
• x*e = y*e
• x = y

### Repeated use of *

Theorem 1.9: For every a in a group (G,*), we define

$\begin{matrix}&a*a*a*a*...*a&\mbox{m times}\\ &*a*a*a*...*a&\mbox{n times}\end{matrix}$

as :

$\begin{matrix} a^m*a^n &=& a^{m+n}\\ &=& a^n*a^m \end{matrix}$

and

$\begin{matrix} (a^n)^m&=& (a^m)^n \end{matrix}$

and

$\begin{matrix} a^{-n}= a^{-1}*a^{-1}*a^{-1}*...*a^{-1} &\mbox{n times} \end{matrix}$

However, when the operation is noted +, we note

$\begin{matrix}&a+a+a+a+...+a&\mbox{m times}\\ &+a+a+a+...+a&\mbox{n times}\end{matrix}$

as :

$\begin{matrix} (n+m)a&=& (m+n)a\\ &=& ma+na\\ &=& na+ma \end{matrix}$

and

$\begin{matrix} -na = (-a)+(-a)+(-a)+(-a)+...+(-a) &\mbox{n times} \end{matrix}$

Where $n,m \in \mathbb{Z}$ (This generalizes the associativity.)

### Groups in which all non-trivial elements have order 2

Theorem 1.10: A group where all non-trivial elements have order 2 is abelian. In other words, if all elements g in a group G satisfy g*g=e, then for any 2 elements g, h in G, g*h=h*g.

• Let g, h be any 2 elements in a group G
• By A1, g*h is also a member of G
• Using the given condition, we know (g*h)*(g*h)=e. Now
• g*(g*h)*(g*h) = g*e
• g*(g*h)*(g*h)*h = g*e*h
• (g*g)*(h*g)*(h*h) = (g*e)*h
• e*(h*g)*e = g*h
• h*g = g*h
• Since the group operation commutes, the group is abelian

## Definitions

Given a group (G, *), if the total number of elements in G is finite, then the group is called a finite group. The order of a group (G,*) is the number of elements in G (for a finite group), or the cardinality of the group if G is not finite. The order of a group G is written as |G| or (less frequently) o(G).

A subset H of G is called a subgroup of a group (G,*) if H satisfies the axioms of a group, using the same operator "*", and restricted to the subset H. Thus if H is a subgroup of (G,*), then (H,*) is also a group, and obeys the above theorems, restricted to H. The order of subgroup H is the number of elements in H.

A proper subgroup of a group G is a subgroup which is not identical to G. A non-trivial subgroup of G is (usually) any proper subgroup of G which contains an element other than e.

Theorem 2.1: If H is a subgroup of (G,*), then the identity eH in H is identical to the identity e in (G,*).

• If h is in H, then h*eH = h; since h must also be in G, h*e = h; so by theorem 1.4, eH = e.

Theorem 2.2: If H is a subgroup of G, and h is an element of H, then the inverse of h in H is identical to the inverse of h in G.

• Let h and k be elements of H, such that h*k = e; since h must also be in G, h*h -1 = e; so by theorem 1.5, k = h -1.

Given a subset S of G, we often want to determine whether or not S is also a subgroup of G. One handy theorem that covers the case for both finite and infinite groups is:

Theorem 2.3: If S is a non-empty subset of G, then S is a subgroup of G if and only if for all a,b in S, a*b -1 is in S.

• If for all a, b in S, a*b -1 is in S, then
• e is in S, since a*a -1 = e is in S.
• for all a in S, e*a -1 = a -1 is in S
• for all a, b in S, a*b = a*(b -1) -1 is in S
• Thus, the axioms of closure, identity, and inverses are satisfied, and associativity is inherited; so S is subgroup.
• Conversely, if S is a subgroup of G, then it obeys the axioms of a group.
• As noted above, the identity in S is identical to the identity e in G.
• By A4, for all b in S, b -1 is in S
• By A1, a*b -1 is in S.

The intersection of two or more subgroups is again a subgroup.

Theorem 2.4: The intersection of any non-empty set of subgroups of a group G is a subgroup.

• Let {Hi} be a set of subgroups of G, and let K = ∩{Hi}.
• e is a member of every Hi by theorem 2.1; so K is not empty.
• If h and k are elements of K, then for all i,
• h and k are in Hi.
• By the previous theorem, h*k -1 is in Hi
• Therefore, h*k -1 is in ∩{Hi}.
• Therefore for all h, k in K, h*k -1 is in K.
• Then by the previous theorem, K=∩{Hi} is a subgroup of G; and in fact K is a subgroup of each Hi.

In a group (G,*), define x0 = e. We write x*x as x² ; and in general, x*x*x*...*x (n times) as xn. Similarly, we write x -n for (x -1)n.

Theorem 2.5: Let a be an element of a group (G,*). Then the set {an: n is an integer} is a subgroup of G.

A subgroup of this type is called a cyclic subgroup; the subgroup of the powers of a is often written as <a>, and we say that a generates <a>.

If there exists a positive integer n such that an=e, then we say the element a has order n in G where n is the smallest n. Sometimes this is written as "o(a)=n".

If S and T are subsets of G, and a is an element of G, we write "a*S" to refer to the subset of G made up of all elements of the form a*s, where s is an element of S; similarly, we write "S*a" to indicate the set of elements of the form s*a. We write S*T for the subset of G made up of elements of the form s*t, where s is an element of S and t is an element of T.

If H is a subgroup of G, then a left coset of H is a set of the form a*H, for some a in G. A right coset is a subset of the form H*a.

Some useful theorems about cosets, stated without proof:

Theorem: If H is a subgroup of G, and x and y are elements of G, then either x*H = y*H, or x*H and y*H have empty intersection.

Theorem: If H is a subgroup of G, every left (right) coset of H in G contains the same number of elements.

Theorem: If H is a subgroup of G, then G is the disjoint union of the left (right) cosets of H.

Theorem: If H is a subgroup of G, then the number of distinct left cosets of H is the same as the number of distinct right cosets of H.

Define the index of a subgroup H of a group G (written "[G:H]") to be the number of distinct left cosets of H in G.

From these theorems, we can deduce the important Lagrange's theorem relating the order of a subgroup to the order of a group:

Lagrange's theorem: If H is a subgroup of G, then |G| = |H|*[G:H].

For finite groups, this also allows us to state:

Lagrange's theorem: If H is a subgroup of a finite group G, then the order of H divides the order of G.

Theorem: If the order of a group G is a prime number, then the group is cyclic.