Definitions

# Elementary group theory

In mathematics, a group <G,*> is defined as a set G and a binary operation on G, called product and denoted by infix "*". Product obeys the following rules (also called axioms). Let a, b, and c be arbitrary elements of G. Then:

• A1, Closure. a*b is in G;
• A2, Associativity. (a*b)*c=a*(b*c);
• A3, Identity. There exists an identity element e in G such that a*e=e*a=a. e, the identity of G, is unique by Theorem 1.4 below;
• A4, Inverse. For each a in G, there exists an inverse element x in G such that a*x=x*a=e. x, the inverse of a, is unique by Theorem 1.5 below.

An abelian group also obeys the additional rule:

Closure is part of the definition of "binary operation," so that A1 is often omitted.

## Elaboration

• Group product "*" 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), 0 usually denotes the identity and -a denotes the inverse of a. The operation is always denoted by something other than * -- often + -- to avoid confusion with multiplication.
• When * is multiplication or a noncommutative operation,a*b is often written ab. 1 usually denotes the identity element, and a -1 usually denotes the inverse of a.
• 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.
• <G,*> is usually pronounced "the group G under *". When affirming that G is a group (for example, in a theorem), we say that "G is a group under *".

## Examples

G={1,-1} is a group under multiplication, because for all elements a, b, c in G:
A1: a*b is an element of G.
A2: (a*b)*c=a*(b*c) can be verified by enumerating all 8 possible (and trivial) cases.
A3: a*1=a. Hence 1 is an identity element.
A4: a-1*a=1. Hence a-1 denotes inverse and 1 is an inverse element.

The integers Z and the real numbers R are groups under addition '+'. For all elements a, b, and c of either Z or R:

A1: Adding any two numbers yields another number of the same kind.
A2: (a+b)+c=a+(b+c).
A3: a+0=a. Hence 0 is an identity element.
A4: -a+a=0. Hence -a denotes inverse and 0 is an inverse element.

The real numbers R are NOT a group under multiplication '*'. For all a, b, and c in R:

A3: 1.
A4: 0*a=0, so 0 has no inverse.
The real numbers without 0, R#, are a group under multiplication '*'.
A1: Multiplying any two elements of R# yields another element of R#.
A2: (a*b)*c=a*(b*c).
A3: a*1=a. Hence 1 denotes identity.
A4: a -1*a=1. Hence a -1 denotes inverse.

## Alternative Axioms

A3 and A4 can be replaced by:

• A3’, left neutral. There exists an e in G such that for all a in G, e*a=a.
• A4’, left inverse. For each a in G, there exists an element x in G such that x*a=e.

Or alternatively by:

• A3’’, right neutral. There exists an e in G such that for all a in G, a*e=a.
• A4’’, right inverse. For each a in G, there exists an element x in G such that a*x=e.

These apparently weaker pairs of axioms are naturally implied by A3 and A4. We will now show that the converse is true.

Theorem: Assuming A1 and A2, A3’ and A4’ imply A3 and A4.

Proof. Let a left neutral element e be given, and a in G. By A4’ there exist an x such that x*a=e.

We show that also a*x=e. Per A4’ there is an y in G with:

$y * \left(a * x\right) = e quad \left(1\right)$

Therefore:


begin{align} e & = y * (a * x) &quad (1)
` & = y * (a * (e * x))       &quad (A3') `
` & = y * (a * ((x * a) * x)) &quad (A4') `
` & = y * (a * (x * (a * x))) &quad (A2)  `
` & = y * ((a * x) * (a * x)) &quad (A2)  `
` & = (y * (a * x)) * (a * x) &quad (A2)  `
` & = e * (a * x)             &quad (1)   `
` & = a * x                   &quad (A3') `
end{align} This establishes A4.


begin{align} a * e & = & a * (x * a) &quad (A4)
`     & = & (a * x) * a &quad (A2) `
`     & = & e * a       &quad (A4) `
end{align} This establishes A3.

Theorem: Assuming A1 and A2, A3’’ and A4’’ imply A3 and A4.

Proof. Similar to the above.

## Basic theorems

### Identity is unique

Theorem 1.4: The identity element of a group <G,*> is unique.

Proof: Suppose that e and f are two identity elements of G. Then


begin{align} e & = & e * f &quad (A3'')
` & = & f     &quad (A3')  `
end{align}

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, eG denotes the identity of the specific group <G,*>.

### Inverses are unique

Theorem 1.5: The inverse of each element in <G,*> is unique.

Proof: Suppose that h and k are two inverses of an element g of G. Then


begin{align} h & = & h * e &quad (A3)
` & = & h * (g * k) &quad (A4) `
` & = & (h * g) * k &quad (A2) `
` & = & (e * k)     &quad (A4) `
` & = & k           &quad (A3) `
end{align}

As a result, we can speak of the inverse of an element a, rather than an inverse. Without ambiguity, for all a in G, we denote by a -1 the unique inverse of a.

### Latin square property

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

Proof. At least one such x surely exists, for if we let x = a -1*b, then x is in G (by A1, closure) and:

• 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.

### Inverting twice gets you back where you started

Theorem 1.6: For all elements a in group G, (a -1) -1=a.

Proof. a -1*a = e. The conclusion follows from Theorem 1.4.

### Inverse of ab

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

Proof. (a*b)*(b -1*a -1) = a*(b*b -1)*a -1 = a*e*a -1 = a*a -1 = e. The conclusion follows from Theorem 1.4.

### Cancellation

Theorem 1.8: For all elements a,x, and y in group G, if a*x=a*y, then x=y; and if x*a=y*a, then x=y.

Proof. 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

### Powers

For $n in mathbb\left\{Z\right\}$ and $a in G$ we define:

a ^ n := begin{cases} underbrace{a*a*cdots*a}_{n mbox{times}}, & mbox{if }n > 0 1, & mbox{if }n = 0 underbrace{a^{-1}*a^{-1}*cdots*a^{-1}}_{-n mbox{times}}, & mbox{if }n < 0 end{cases}

Theorem 1.9: For all a in group <G,*>, $n, m in mathbb\left\{Z\right\}$:


begin{matrix} a^m*a^n &=& a^{m+n} (a^m)^n &=& a^{m*n} end{matrix}

Similarly if G is written in additive notation, we have:


n * a := begin{cases} underbrace{a+a+cdots+a}_{n mbox{times}}, & mbox{if }n > 0 0, & mbox{if }n = 0 underbrace{(-a)+(-a)+cdots+(-a)}_{-n mbox{times}}, & mbox{if }n < 0 end{cases}

and:


begin{matrix} (m*a) +(n*a) &=& (m+n)*a m*(n*a) &=& (m*n)*a end{matrix}

## Order

### Of a group element

The order of an element a in a group G is the least positive integer n such that an = e. Sometimes this is written "o(a)=n". n can be infinite.

Theorem 1.10: A group whose nontrivial elements all have order 2 is abelian. In other words, if all elements g in a group G g*g=e is the case, then for all elements a,b in G, a*b=b*a.

Proof. Let a, h be any 2 elements in the group G. By A1, a*h is also a member of G. Using the given condition, we know that (a*h)*(a*h)=e. Hence:

• a*(a*b)*(a*b) = a*e
• a*(a*b)*(a*b)*b = a*e*b
• (a*a)*(b*a)*(b*b) = (a*e)*b
• e*(b*a)*e = a*b
• b*a = a*b.

Since the group operation * commutes, the group is abelian

### Of a group

The order of the group G, usually denoted by |G| or occasionally by o(G), is the number of elements in the set G, in which case <G,*> is a finite group. If G is an infinite set, then the group <G,*> has order equal to the cardinality of G, and is an infinite group.

## Subgroups

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,*).

Proof. 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.

Proof. 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. A handy theorem valid for both infinite and finite 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.

Proof. 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.

Proof. 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.

Given a group <G,*>, define x*x as x², x*x*x*...*x (n times) as xn, and define x0 = e. Similarly, let x -n for (x -1)n. Then we have:

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.

Proof. 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>.

## Cosets

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.

If H is a subgroup of G, the following useful theorems, stated without proof, hold for all cosets:

• And x and y are elements of G, then either x*H = y*H, or x*H and y*H have empty intersection.
• Every left (right) coset of H in G contains the same number of elements.
• G is the disjoint union of the left (right) cosets of H.
• Then the number of distinct left cosets of H equals 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:

• : If H is a subgroup of G, then |G| = |H|*[G:H].

For finite groups, this can be restated as:

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