Elementary group theory

2008/9 Schools Wikipedia Selection. Related subjects: Mathematics

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:

A5, Commutativity. a*b = b*a.

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

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

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, e_G 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 $n \in \mathbb{Z}$ 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,*⟩, $n, m \in \mathbb{Z}$ :

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

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 e_H in H is identical to the identity e in (G,*).

Proof. If h is in H, then h*e_H = h; since h must also be in G, h*e = h; so by theorem 1.4, e_H = 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 {H_i} be a set of subgroups of G, and let K = ∩{H_i}. e is a member of every H_i 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 H_i.
By the previous theorem, h*k^-1 is in H_i
Therefore, h*k^-1 is in ∩{H_i}.

Therefore for all h, k in K, h*k^-1 is in K. Then by the previous theorem, K=∩{H_i} is a subgroup of G; and in fact K is a subgroup of each H_i.

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:

Lagrange's theorem: 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.

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