1
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 that satisfies the following axioms
1. : Closure: For all , .
2. : Associativity: For all ,
3. : Identity: For any , there exists an where .
4. : Inverse: For each , there exists an element where .
Examples of Groups
Example 1: Show that the integers is a group under addition.
Solution: For , we show it satisfies the 4 properties for a group.
1. : For , it is known that Z is closed under + , that is .
2. : (Z is known to be associative under +)
3. : For each , there exists an identity zero given by were
(0 is the known additive identity element in the integers)
4. : For each , there exists an where
(Each element in Z has an additive inverse obtained by
negating the element)
█
Example 2: Show that the non-zero rationals is a group under multiplication.
Solution: For , we show it satisfies the 4 properties for a group.
1. : For , it is known that is closed under , that is .
1. : ( is known to be associative under +)
2. : For each , there exists an identity zero given by were
(1 is the known multiplicative identity element in the non-zero
rationals)
3. : For each , there exists an element where
(Each element in Z has a multiplcative inverse obtained by
taking the reciprocal of the element)
█
Example 3: Determine why the set under addition is not a group.
Solution:
█
Example 4: Determine whether the binary operation defined on Z where represents that of a group.
Solution:
█
Example 5: Determine whether the binary operation defined on by is a group.
Solution:
█
Definition 1: A group G is abelian if its binary operation is commutative.
Examples of abelian groups:
· is abelian since integer addition is known to be commutative, that is, for all .
· where is abelian since for , .
Definition 1: The set is defined to be the set of all possible integer multiples of the integer k.
Example 6: Determine and .
Solution:
█
Example 7: Determine if is an abelian group under +.
Solution: For , we show it satisfies the 4 properties for a group.
1. : Closure. Let . Then and for . Then
. Hence, is closed under + .
2. : (a, b, c are just integers, and integer arithmetic is known
to be associative under +)
3. : For each , there exists an identity zero given by were
(0 is the known additive identity element in the integers under +)
4. : For each , for. Hence, there exists a for
where (Each element in kZ has an additive inverse
obtained by negating the element)
Thus, is a group under +. Also, for all since a, b are just integers, and integer arithmetic is known to be commutative under +.
Hence, gives an abelian group under +
█
Example 8: Determine if the set of matrices with a non-zero determinant is an abelian group under matrix multiplication.
Solution:
█
The Group
Recall that the set of integers modulo m is defined by the set .
Example 9: Determine if is an abelian group under the addition operator.
Solution: For , we show it satisfies the 4 properties for a group. To do this, let the binary operation denote the addition modulo m (for example, can be thought of computationally as finding the integer remainder of when divided by m.
Hence, we check the 4 group properties.
1. : Closure. Let . Then, it is straightforward to see that and the closure property holds.
2. : Associativity. Using the division algorithm, we can say that
Adding these two equations gives the equation
Canceling the from both sides of the equation gives the equality
Similarly, if we can use the division algorithm to compute
Adding these two equations gives the equality
.
Hence, we have two different equations representing when this number is divided by m, namely
Recall that division algorithm says that when dividing two numbers, there is only one possible quotient and remainder. Hence, this says that
and hence the associativity property holds.
3. : For each , it is clear that the element serves as the identity where
.
4. : For each , would represent the inverse for a since it is straight
forward to see that .
Thus, is a group under . Also, since integer addition in general is commutative, it is straight forward to see that for all . Thus, is an abelian group.
█
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 , the for all , if , then and if , then .
Proof:
█
2. Unique Solution to Linear Equations: If G is a group with binary operation , then for , the linear equations and have unique solutions .
Proof:
█
3. Uniqueness of Identity Element and Inverse: In a group G, there is only one element such that for each . Likewise, there is one element such that for each .
Proof: Suppose that represent two identity elements of the group G. Then, and are identity elements of every element of G, including each other. Hence, it
follows that
Suppose and are both inverses of an arbitrary element . Then
█
4. If G is a group with binary operation , the for all ,
Proof:
█
5. For any element in a group G with binary operation , .
Proof: For , and . Since and the inverse of any element is unique (fact 3), it follows that █
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 , , and if we are working in the real number system.
Solution:
█
Note that the real numbers fail to be a group under multiplication since 0 fails to have a multiplicative inverse since does not exist. However, if we consider the non-zero real numbers , 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 that have multiplicative inverses under multiplication.
Solution:
█
For the set , 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 , then has a multiplicative inverse in .
Definition 3: The set of elements in with multiplicative inverses, denoted by , is called the set of units of . That is,
Fact: The set of units represent an abelian group under multiplication.
Note that is closer under multiplication since if , since . is associative since it is a subset of and all elements of are associative under multiplication. The identity element is and all elements in have multiplicative inverses. abelian since it is a subset of and all elements of are commutative under multiplication.
Example 11: Determine the group of units of and of
Solution:
█
This leads to the following important fact.
Fact: If p is a prime number, the set of non-zero elements is a group under multiplication.
The fact follows that since the only divisors of a prime p are 1 and itself, if , then , which implies . This says that all elements of have multiplicative inverses.
Exercises
1. Determine whether the binary operation defined on the following sets represent a group. If not a group, state a reason why.
a. Let be defined on by letting .
b. Let be defined on by letting .
c. Let be defined on by letting .
d. Let be defined on by letting .
e. Let be defined on by letting .
f. Let be defined on by letting .
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 diagonal matrices under matrix addition.
b. All diagonal matrices under matrix multiplication.
c. All diagonal matrices with no zero diagonal entry under under matrix multiplication.
d. All 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 on S by
a. Show that is a group.
b. Find the solution of the equation in S.
4. If in a group, prove that .
5. For a group G, if and , prove that .
6. If is a binary operation on a set S, an element is an idempotent for if . Prove that a group has exactly one idempotent element.
7. Show that every group G with identity e such that for all is abelian. Hint: Consider .
8. Show that if for a and b in a group G, then .
9. Let G be a group and let . Show that if and only if .
10. Let G be a group and suppose for . Show that also.
11. Show that if G is a finite group with identity e and with an even number of elements, then there is a in G such that .
12. List the elements of the following group of units.
a.
b.
c.
d.
e.
f.