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Groups

Practice HW # 1-12 p. 18 at the end of the notes

In this section, we discuss the basics of groups. As we will see, the most basic number systems that we are accustomed to working with are examples of groups.

Groups

A group G is a non-empty set that under a binary operation [pic] that satisfies the following axioms

1. [pic]: Closure: For all [pic], [pic].

2. [pic]: Associativity: For all [pic], [pic]

3. [pic]: Identity: For any [pic], there exists an [pic] where [pic].

4. [pic]: Inverse: For each [pic], there exists an element [pic] where [pic].

Examples of Groups

Example 1: Show that the integers [pic] is a group under addition.

Solution: For [pic], we show it satisfies the 4 properties for a group.

1. [pic]: For [pic], it is known that Z is closed under + , that is [pic].

2. [pic]: [pic] (Z is known to be associative under +)

3. [pic]: For each [pic], there exists an identity zero given by [pic] were

[pic] (0 is the known additive identity element in the integers)

4. [pic]: For each [pic], there exists an [pic] where

[pic] (Each element in Z has an additive inverse obtained by

negating the element)

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Example 2: Show that the non-zero rationals [pic] is a group under multiplication.

Solution: For [pic], we show it satisfies the 4 properties for a group.

1. [pic]: For [pic], it is known that [pic] is closed under [pic] , that is [pic].

1. [pic]: [pic] ([pic] is known to be associative under +)

2. [pic]: For each [pic], there exists an identity zero given by [pic] were

[pic] (1 is the known multiplicative identity element in the non-zero

rationals)

3. [pic]: For each [pic], there exists an element [pic] where

[pic] (Each element in Z has a multiplcative inverse obtained by

taking the reciprocal of the element)

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Example 3: Determine why the set [pic]under addition is not a group.

Solution:

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Example 4: Determine whether the binary operation [pic] defined on Z where [pic] represents that of a group.

Solution:

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Example 5: Determine whether the binary operation [pic] defined on [pic] by [pic] is a group.

Solution:

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Definition 1: A group G is abelian if its binary operation is commutative.

Examples of abelian groups:

• [pic] is abelian since integer addition is known to be commutative, that is, [pic] for all [pic].

• [pic] where [pic]is abelian since for [pic], [pic].

Definition 1: The set [pic] is defined to be the set of all possible integer multiples of the integer k.

Example 6: Determine [pic] and [pic].

Solution:

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Example 7: Determine if [pic] is an abelian group under +.

Solution: For [pic], we show it satisfies the 4 properties for a group.

1. [pic]: Closure. Let [pic]. Then [pic] and [pic] for [pic]. Then

[pic]. Hence, [pic] is closed under + .

2. [pic]: [pic] (a, b, c are just integers, and integer arithmetic is known

to be associative under +)

3. [pic]: For each [pic], there exists an identity zero given by [pic] were

[pic] (0 is the known additive identity element in the integers under +)

4. [pic]: For each [pic], [pic] for[pic]. Hence, there exists a [pic] for

[pic] where [pic] (Each element in kZ has an additive inverse

obtained by negating the element)

Thus, [pic] is a group under +. Also, [pic] for all [pic] since a, b are just integers, and integer arithmetic is known to be commutative under +.

Hence, [pic] gives an abelian group under +

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Example 8: Determine if the set of [pic] matrices with a non-zero determinant is an abelian group under matrix multiplication.

Solution:

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The Group [pic]

Recall that the set of integers modulo m is defined by the set [pic].

Example 9: Determine if [pic] is an abelian group under the addition operator.

Solution: For [pic], we show it satisfies the 4 properties for a group. To do this, let the binary operation [pic] denote the addition modulo m (for example, [pic] can be thought of computationally as finding the integer remainder of [pic] when divided by m.

Hence, we check the 4 group properties.

1. [pic]: Closure. Let [pic]. Then, it is straightforward to see that [pic] and the closure property holds.

2. [pic]: Associativity. Using the division algorithm, we can say that

[pic]

Adding these two equations gives the equation

[pic]

Canceling the [pic] from both sides of the equation gives the equality

[pic]

Similarly, if we can use the division algorithm to compute

[pic]

Adding these two equations gives the equality

[pic].

Hence, we have two different equations representing [pic] when this number is divided by m, namely

[pic]

[pic]

Recall that division algorithm says that when dividing two numbers, there is only one possible quotient and remainder. Hence, this says that

[pic]

and hence the associativity property holds.

3. [pic]: For each [pic], it is clear that the element [pic] serves as the identity where

[pic].

4. [pic]: For each [pic], [pic] would represent the inverse for a since it is straight

forward to see that [pic].

Thus, [pic] is a group under [pic]. Also, since integer addition in general is commutative, it is straight forward to see that [pic] for all [pic]. Thus, [pic] is an abelian group.

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The next question we want to examine more closely has to do with sets defined under multiplication and when sets defined under multiplication are groups and when not.

Properties of Groups

1. Left and Right Commutative Laws: If G is a group with binary operation [pic], the for all [pic], if [pic], then [pic] and if [pic], then [pic].

Proof:

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2. Unique Solution to Linear Equations: If G is a group with binary operation [pic], then for [pic], the linear equations [pic]and [pic] have unique solutions [pic].

Proof:

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3. Uniqueness of Identity Element and Inverse: In a group G, there is only one element [pic] such that [pic] for each [pic]. Likewise, there is one element [pic] such that [pic] for each [pic].

Proof: Suppose that [pic]represent two identity elements of the group G. Then, [pic]and [pic] are identity elements of every element of G, including each other. Hence, it

follows that

[pic]

Suppose [pic] and [pic] are both inverses of an arbitrary element [pic]. Then

[pic]

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4. If G is a group with binary operation [pic], the for all [pic],

[pic]

Proof:

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5. For any element [pic]in a group G with binary operation [pic], [pic].

Proof: For [pic], [pic] and [pic]. Since [pic] and the inverse of any element is unique (fact 3), it follows that [pic] █

Multiplicative Inverses

In the real number system, every non-zero number has a multiplicative inverse – the number you must multiply to a given number to get 1. In other words, every non-zero real number is a unit.

Example 10: Fill in the ( ) for [pic], [pic], and [pic] if we are working in the real number system.

Solution:

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Note that the real numbers [pic] fail to be a group under multiplication since 0 fails to have a multiplicative inverse since [pic] does not exist. However, if we consider the non-zero real numbers [pic], we will have a group under multiplication.

For some sets defined over multiplication, multiplicative inverses in most cases do not exist.

Example 11: Name the elements in the set of integers [pic] that have multiplicative inverses under multiplication.

Solution:

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For the set [pic], multiplicative inverses exist depending on a given condition, which we explain next.

Definition 2: The greatest common divisor of two numbers, denoted as gcd(a,b), is the largest number that divides a and b evenly with no remainder.

For example, gcd(10, 20). = 10 and gcd(72, 108) = 36.

The condition for which a multiplicative inverse exists is given by the following theorem, which for now we state without proof.

Theorem 1: If the [pic], then [pic] has a multiplicative inverse in [pic].

Definition 3: The set of elements in [pic] with multiplicative inverses, denoted by [pic], is called the set of units of [pic]. That is,

[pic]

Fact: The set of units [pic] represent an abelian group under multiplication.

Note that [pic] is closer under multiplication since if [pic], [pic] since [pic]. [pic] is associative since it is a subset of [pic] and all elements of [pic] are associative under multiplication. The identity element is [pic] and all elements in [pic] have multiplicative inverses. [pic] abelian since it is a subset of [pic] and all elements of [pic] are commutative under multiplication.

Example 11: Determine the group of units [pic] of [pic] and [pic] of [pic]

Solution:

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This leads to the following important fact.

Fact: If p is a prime number, the set of non-zero elements [pic] is a group under multiplication.

The fact follows that since the only divisors of a prime p are 1 and itself, if [pic], then [pic], which implies [pic]. This says that all elements of [pic] have multiplicative inverses.

Exercises

1. Determine whether the binary operation [pic] defined on the following sets represent a group. If not a group, state a reason why.

a. Let [pic]be defined on [pic] by letting [pic].

b. Let [pic]be defined on [pic] by letting [pic].

c. Let [pic]be defined on [pic] by letting [pic].

d. Let [pic]be defined on [pic] by letting [pic].

e. Let [pic]be defined on [pic] by letting [pic].

f. Let [pic]be defined on [pic] by letting [pic].

2. A diagonal matrix is a square matrix whose only nonzero entries lie on the main diagonal, from the upper left to the lower right corner. Determine if the following set of matrices under the specified operation is a group.

a. All [pic] diagonal matrices under matrix addition.

b. All [pic] diagonal matrices under matrix multiplication.

c. All [pic] diagonal matrices with no zero diagonal entry under under matrix multiplication.

d. All [pic] diagonal matrices with all diagonal entries 1 or -1 under under matrix multiplication.

3. Let S be the set of all real numbers except -1. Define [pic] on S by

[pic]

a. Show that [pic] is a group.

b. Find the solution of the equation [pic] in S.

4. If [pic] in a group, prove that [pic].

5. For a group G, if [pic] and [pic], prove that [pic].

6. If [pic] is a binary operation on a set S, an element [pic]is an idempotent for [pic]if [pic]. Prove that a group has exactly one idempotent element.

7. Show that every group G with identity e such that [pic] for all [pic] is abelian. Hint: Consider [pic].

8. Show that if [pic] for a and b in a group G, then [pic].

9. Let G be a group and let [pic]. Show that [pic] if and only if [pic].

10. Let G be a group and suppose [pic] for [pic]. Show that [pic] also.

11. Show that if G is a finite group with identity e and with an even number of elements, then there is a [pic] in G such that [pic].

12. List the elements of the following group of units.

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

f. [pic]

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