Introduction to group theory

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Group theory is the study of algebraic structures called groups. This introduction will rely heavily on set theory and modular arithmetic as well. Later on it will require an understanding of mathematical induction, functions, bijections, and partitions. Lessons may utilize matricies and complex numbers as well.

After completing this section move on to Introduction to group theory/Part 2 Subgroups and cyclic groups.

Introduction[edit]

What is a group? A more rigorous definition will come shortly but to give a very rough idea of a group it is a set and a operation. For example the integers under addition form a group, notated as (, +) or simply , where the operation is assumed to be addition.

Some more examples of groups are

  1. The real numbers under addition: (, +) or simply
  2. The rational numbers under addition: (, +) or simply
  3. The non-zero real numbers under multiplication: (*, ×)
  4. The set of 2x2 matricies with integer entries under matrix addition: (M2(), +)

All of these structures have things in common; they are all integral to being groups. They also have things in common that aren't necessary to groups. Let's examine some of these similarities.

Closure Under the Operation[edit]

All of these groups have a closed binary operation. For example in (, +) any two integers added together will be another integer. In other words if n,m ∈ then (n+m)∈.

In general for (G, *) to be a group where G is a set and * is a binary operation, if a,b are in G then (a*b) is also in G. This is called closure. Notice that all of the groups in the above examples are closed under their respective operations.

Associativity[edit]

With the integers under addition

With the non-zero real numbers under multiplication

This is called associativity and is required for a structure to be a group. In general if (G,*) is a group and a,b,c∈G then

Identity[edit]

When we look at there's something special about the element 0. Notice that for any integer m

Zero is the only element in this group with this property and it's called the identity of the group.

Zero is also the identity in the groups , and .

In * the element is the identity as

for all a in .

In general if (G,*) is a group then there exists an identity element e in G such that for any g in G

This element is called the identity of G or eG.

Inverses[edit]

In if is an integer consider

It would then follow that and in fact is an integer as well.

In * if r is a non-zero real number then

has a solution. Further and x is also a non-zero real number.

In general if (G, *) is a group with identity and is an element of G then there exists an element also in G such that

.

Note that at this point is purely notational. If we are looking at the group of integers under addition then means since

.

It does not mean in this group.

Possible Misconceptions[edit]

In all of the above examples the underlying set of the groups are infinite, but groups need not be infinite. Note that with the requirement of an identity element the underlying set cannot be the empty set.

All of the groups above are commutative. That is that . This is not true of all groups in general. Groups that are commutative are called Abelian Groups.

Non-groups[edit]

To solidify our understanding let's look at some structures that aren't groups.

Firstly ({0,1,2,3},+) is not a group as and is not in {0,1,2,3} and this set is not closed under our operation.

Consider under addition. This set is closed but it doesn't have inverses therefore it is not a group.

Consider the set of all matricies under addition. This is not a group because not all matricies can be added. Consider for example a 2x2 matrix and a 3x3 matrix.

Consider (, *). This is not a group because 0 doesn't have an inverse and since , there is no identity.

Definition[edit]

A set G under the closed binary operation * is a group denoted (G,*) or simply G iff

  1. G under * is associative:
  2. G under * has an identity element: such that
  3. Each element in G has an inverse under *: such that where is the identity in G.

Notational Notes[edit]

Since groups are associative it is common place to drop the parentheses when one is working with something shown to be a group. If a structure has yet to be shown to be associative do not drop the parentheses when working with elements of it. Do not however drop parentheses when working with inverses. For example and are not necessarily the same. Note that is assumed to mean .

Since groups only have one operation is is usually dropped much like multiplication in elementary algebra. For example:

becomes .

Dropping both the parentheses and the operation symbol leads to long strings of elements being unambiguous. For example any interpretation of is equivalent. I.e.

In most groups is assumed to be the identity and is used in arbitrary groups where the identity is unknown.

When strings of the same element are being multiplied we use exponent notation to represent it. For example

Do note that we must be careful not to assume elements commute. Thus

but can be simplified no further.

In abelian groups (commutative groups) and later on in the study of Rings additive notation can be used in place of multiplicative. For example

becomes .

Multiplying[edit]

Note that "=" is an equivalence relation and thus we can substitute. For example in a group G suppose such that . Then by closure , and by reflexivity . We may substitute to arrive at . Thus

This is called multiplying on the right by . Similarly

is called multiplying on the left.

Advice[edit]

Now we may begin to play with some equations. Moving on it is best to try to "forget" our assumptions about algebra we have learned from our elementary courses and only use what is explicitly proven.

Theorems From the Definition[edit]

These are important theorems that follow directly from the definition of a group. Attempt the proofs yourself before looking at the solutions.

Uniqueness of the Identity Element[edit]

An important theorem to begin with is the uniqueness of the identity. More precisely stated: Let G be a group. If

such that

then

Proof

Cancellation[edit]

This theorem lets us cancel elements exactly opposite of how we multiply them.

Right Cancellation Theorem: .

Right Cancellation Proof

Left cancellation is similarly proven. Theorem: .

Uniqueness of Inverses[edit]

This theorem states that each element has only one inverse. Theorem: Let G be a group. Then if such that and are both inverses of then .

Proof

Socks and Shoes[edit]

This theorem is a way to distribute inverses.

Theorem: For group elements and ,

.

Induction can be used to prove the more powerful socks and shoes theorem.

Theorem: For groups elements

.

Proof

Integer Modulo Groups[edit]

Note that = {0,1,2,...,n-1}. It happens that (,+mod(n)) is a group and is written for short.

Arithmetic Examples[edit]

In

In

Dihedral Groups[edit]

The dihedral groups arise from looking at the symmetries of regular polygons.

Cayley Tables[edit]

A cayley table is a table that displays the products of elements of the group under the operation.

For example, the Cayley table of is:

+ 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1

Homework[edit]

Attempt the proofs and problems on your own before looking at the solutions.

Problem 1.

Prove that cross cancellation implies commutativity. That is assume

and show Solution