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Elementary group theory

Encyclopedia : E : EL : ELE : 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:

Contents

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.

Identity relations are commutative

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

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. 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. 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.

The inverse of ab

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

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.

Repeated use of *

Theorem 1.9: For every a in a group, aman = am+n = anam and (am)n = (an)m = anm. (This generalizes the associativity.)

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,*).

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. 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.

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.

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 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.

References

 

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