Groups

Definition

A group G, is a set G, closed under a binary operation , such that the following axioms are satisfied:

1) Associativity of :

For all a, b, cG, we have (a b) c = a (b c).

2) Identity element e for :

There is an element e in G such that for all x G, e x = x e = x

3) Inverse a’ of a:

For each a G, there is an element a’ in G such that

a a’ = a’ a = e.

Examples

DefinitionA group G is abelian if its binary operation is commutative.Example:• The set Z+ under addition is not a group. since there is no identity element for + in Z+.

• The set of all nonnegative integers under addition is not a group since there is no inverse for 2.

• The set Z, Q, R, and C under addition are abelian groups.

• The set Z+ under multiplication is not a group since there is no inverse of 3.

Examples

• The sets Q+ and R+, Q*, R*, C* under multiplication are abelian groups.

Example: Let be defined on Q+ by a b=ab/2. Determine Q+ under such a binary operation is a group.

It is a group since it satisfies the three properties of a group:(1) (a b) c=(ab/2) c=(abc)/4, and a (b c)=a (bc/2)=(abc)/4.

Thus is associative.(2) a e=ae/2=a implies e=2. We have 2 a=a 2=a for all a Q+. So 2

is an identity element for .(3) a a’=aa’/2=e=2 implies a’=4/a. We have a 4/a=4/a a=2. So

a’=4/a is an inverse for a.Hence Q+ with the operation is a group. #

Elementary Properties of Groups

Theorem 4.15

If G is a group with binary operation , then the left and right cancellation laws hold in G.

that is, a b=a c implies b=c, and b a=c a implies b=c for all a, b, c G.

Proof: Suppose a * b = a * c. Then there exists an inverse of a’ to a. Apply this inverse on the left,

a’ * (a * b) = a’ *(a * c)By the associatively law, (a’ * a ) * b = (a’ * a) * cSince a’ is the inverse of a, a’ * a =e, we have

e * b = e * cBy the definition of e,

b = cSimilarly for the right cancellation. #

Theorem 4.16

If G is a group with binary operation , and if a and b are any elements of G, then the linear equations a x=b and y a=b have unique solutions x and y in G.

Proof: First we show the existence of at least one solution by just computing that a’ b is a solution of a x=b. Note that

a * (a’ * b) = (a *a’) * b, associative law, =e * b, definition of a’, =b, property of e.

Thus x= a’ b is a solution a x=b. In a similar fashion, y=b a’ is a solution of y a=b.

To show uniqueness of y, we assume that we have two solutions, y1 and y2, so that y1 a=b and y2 a=b. Then y1 a=y2 a, and by Theorem 4.15, y1=y2. The uniqueness of x follows similarly. #

Theorem 4.17

In a group G with binary operation , there is only one element e in G such that

e x = x e = x

for all x G. Likewise for each a G, there is only one element a’ in G such that

a’ a = a a’ = e

In summary, the identity element and inverse of each element are unique in a group.

Proof: We’ve shown the uniqueness of an identity element for any binary structure in section 3.

Uniqueness of an inverse

Suppose that a G has an inverses a’ and a’’ so that a’ a = a a’ = e and a’’ a = a a’’ = e. Then

a a’’= a a’ = e

And, by Theorem 4. 15,

a’’=a’

So the inverse of a in a group is unique.

#

Corollary

Let G be a group. For all a, b G, we have (a b)’ = b’ a’.

Proof: in a group G, we have

(a b) (b’ a’) = a (b b’) a’ = (a e) a’= a a’=e.By theorem 4.17, b’ a’ is the unique inverse of a b. That is, (a b)’ = b’ a’.

#

Group Table

Every group table is a Latin square; that is, each element of the group appears exactly once in each row and each column.

Proof: On the contrary, suppose x appears in a row labeled with a twice. Say x=a b and x=a c. Then cancellation gives b=c. This contradicts the fact that we use distinct elements to label the columns.

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Finite Groups

One-element Group{e} with the binary operation defined by e e=e

Two-element GroupExample: {e, a}, try to find a table for a binary operation on {e, a} that

gives a group structure on {e, a}. e a e e a a a x

Since every element can occur exactly once in each row and each column, we have x =e .

Three-element group

Three-element group

Example: {e, a, b}, try to find a table for a binary operation on {e, a, b} that gives a group structure on {e, a, b}.

e a b

e e a b

a a x y

b b z w

Since every element can occur exactly once in each row and each column, we have x=b, y=e, z= e, w=a.

Note: There is only one group of three elements, up to isomorphism.