Group Theory

Definitions

Algebraic System

Set of elements with a rule for combining elements of a set.

Group

A nonempty set G is a group if in G there is an operation * so that:

Abelian Group

A group is said to be abelian if a*b = b*a for every a,b in that group.

Subgroup

A nonempty subset of a group is said to be a subgroup if itself forms a group under the operation of the group.

Order

If G is finite, then the order of a is the least positive integer n such that a^n = e

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Examples

Algebraic System:

A group is an example of an algebraic system.

Example of a non-group: Natural Numbers under addition

The set of natural numbers do not form a group under addition because it has no identity. The only real number that could act as an identity to the natural numbers is 0 because if a in the set of natural numbers then a + 0 = 0 +a = a but 0 is not a natural number. Since there is no identity there can't be an inverse.

Example of a group: Integers under addition

The set of all intergers on the other hand does form a group under addition.

By the basic properties of addition, the set of integers is closed and associative under addition.

0 is in the set of integers and acts as the identity. Let a be in the set of integers, a + 0 = a + (0 + 0) = 0, so a + 0 = 0. Similiarly, 0 + a = (0 +0) + a = 0

Let a and b be in the set of integers where b = -a, a + b = a + (-a) = a - a = 0 so for every a there exists an inverse.

Therefore the set of integers forms a group.

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Properties of Groups

If G is a group, then:

1. Its identity is unuque

Proof: Suppose the identity isn't unique. Suppose that e and e* are identities. Then ea = ae = a and e*a = ae* = a. So by substitution, ea = e*a and ae = ae*, so e = e*. Thus the identity is unique.

2. Every a in G has a unique inverse in G.

Proof: Suppose Not. Suppose that a in G has two inverses, a^(-1) and a^(-1)*. Then aa^(-1) = a^(-1)a = e and aa^(-1)* = a(-1)*a = e. So aa^(-1) = aa^(-1)* and a^(-1)a = a^(-1)*a. Therefore a^(-1) = a^(-1)*.

3. If a is in G then (a^(-1))^(-1) = a

Proof: Let a be in G. Since G is a group, a^(-1) * a = e. Since a^(-1) is in G, a^(-1) * (a^(-1)^-1) = e. So again by substitution, a^(-1) * a = a^(-1) * (a^(-1)^-1). Thus a = (a^(-1)^-1).

4. If a,b is in G then (ab)^(-1) = b^(-1)a^(-1)

Proof: (ab)(b^(-1)a^(-1)) = a(bb^(-1))a^(-1) = aea^(-1) = (ae)a^(-1) = aa^(-1) = e. So (ab)^(-1) = b^(-1)a^(-1).

5. Let a,b,c be in G. Then:

i) if ab = ac then b = c

Proof: Let a,b,c be in G and let ab = ac. Then a^(-1)ab = a^(-1)ac, so eb = ec, thus b = c

and

ii) if ba = ca then b = c.

Proof: Let a,b,c be in G and let ba = ca. Then baa^(-1) = caa^(-1) , so be=ce, thus b = c.

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