Unit 3
Algebra
Class 12 Mathematics
Elementary Group Theory
Class 12 Mathematics – Elementary Group Theory Notes PDF
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Contents / Quick Navigation
NEB / CDC Focus
This chapter introduces a binary operation as a rule that combines two elements of a set to produce another element of the same set. It then develops the concept of a group, finite and infinite groups, uniqueness of identity and inverse, the cancellation laws, and Abelian groups. The treatment below stays within that Class 12 syllabus scope.
1. Introduction
Elementary group theory studies a set together with an operation that satisfies a small collection of algebraic rules. The central idea is not the kind of objects in the set but how the operation behaves.
A group is written as (G, *), where G is a non-empty set and * is a binary operation on G satisfying the group axioms.
2. Binary Operation
Let S be a non-empty set. A binary operation * on S is a rule that assigns to every ordered pair (a,b)∈S×S a unique element a*b∈S.
* : S × S → S
2.1 Conditions Hidden in the Definition
- Two inputs: the operation acts on an ordered pair (a,b).
- Defined for every pair: no pair from S×S may be left undefined.
- Unique output: each ordered pair has one well-defined result.
- Closure: the output belongs to S.
Addition is a binary operation on the integers ℤ because for any a,b∈ℤ, the sum a+b is again an integer.
Division is not a binary operation on ℤ. For example, 1÷2=1/2∉ℤ, and division by zero is not defined.
3. Binary Operations on Sets of Integers
When a rule is defined on integers, first check whether it always gives an integer. That is the closure test.
| Operation on ℤ | Closed? | Reason |
|---|---|---|
| a*b=a+b | Yes | Sum of two integers is an integer. |
| a*b=a-b | Yes | Difference of two integers is an integer. |
| a*b=ab | Yes | Product of two integers is an integer. |
| a*b=a+b+3 | Yes | For integers a and b, the result is an integer. |
| a*b=(a+b)/2 | No | For a=1, b=2, the result is 3/2, not an integer. |
| a*b=a/b | No | Not closed and not defined when b=0. |
On ℤ, define a*b=a+b-ab.
Since sums and products of integers are integers, a+b-ab is an integer for every a,b∈ℤ. Hence * is closed on ℤ.
4. Important Properties of Binary Operations
4.1 Closure
For all a,b∈S, a*b∈S.
4.2 Associative Property
Associativity changes only the grouping, not the order of the elements.
4.3 Commutative Property
Commutativity changes the order of the two elements. It is not required for every group; it is the extra condition that makes a group Abelian.
4.4 Identity Element
An element e∈S is an identity if a*e=e*a=a for every a∈S.
4.5 Inverse Element
If e is the identity, an element b∈S is an inverse of a if a*b=b*a=e. It is usually written a-1.
5. Definition of a Group
A non-empty set G with a binary operation * is called a group if:
- Closure: a*b∈G for all a,b∈G.
- Associativity: (a*b)*c=a*(b*c) for all a,b,c∈G.
- Identity: there exists e∈G such that a*e=e*a=a for every a∈G.
- Inverse: for every a∈G, there exists a-1∈G such that a*a-1=a-1*a=e.
- Closure: a+b∈ℤ.
- Associativity: (a+b)+c=a+(b+c).
- Identity: 0.
- Inverse of a: -a.
Therefore (ℤ,+) is a group. It is also Abelian because a+b=b+a.
Although integer multiplication is closed and associative and has identity 1, most integers have no multiplicative inverse in ℤ. For example, the inverse of 2 would be 1/2, which is not an integer. Thus (ℤ,×) is not a group.
6. How to Test Whether a Structure Is a Group
- Closure: take arbitrary a,b∈G and show a*b∈G.
- Associativity: compare (a*b)*c with a*(b*c).
- Identity: solve a*e=e*a=a for e; it must belong to G and work for every a.
- Inverse: solve a*b=b*a=e; the inverse must belong to G for every a.
- Abelian check: after proving it is a group, check whether a*b=b*a.
7. Finite and Infinite Groups
A group containing a finite number of elements is called a finite group. The number of elements is called the order of the group.
G={1,-1} under ordinary multiplication is a finite group of order 2.
A group with infinitely many elements is called an infinite group.
(ℤ,+) is an infinite group.
8. Uniqueness of Identity Element
The identity element of a group is unique.
Proof
Suppose a group G has two identity elements e and f.
- Since e is an identity, e*f=f.
- Since f is an identity, e*f=e.
Therefore e=f. Hence the identity is unique.
9. Uniqueness of Inverse
Every element of a group has a unique inverse.
Proof
Let a∈G. Suppose b and c are both inverses of a. Then:
a*b=b*a=e, a*c=c*a=e.
Now, using the identity and associativity:
b=b*e=b*(a*c)=(b*a)*c=e*c=c.
Thus b=c. Hence the inverse of each element is unique.
(a-1)-1=a.
Also, in any group, (a*b)-1=b-1*a-1. Notice the reversed order.
10. Cancellation Law
In a group (G,*):
a*b=a*c ⇒ b=c (left cancellation) b*a=c*a ⇒ b=c (right cancellation)10.1 Proof of Left Cancellation
Given a*b=a*c, operate on both sides from the left by a-1:
a-1*(a*b)=a-1*(a*c).
By associativity:
(a-1*a)*b=(a-1*a)*c ⇒ e*b=e*c ⇒ b=c.
10.2 Proof of Right Cancellation
Given b*a=c*a, multiply from the right by a-1:
(b*a)*a-1=(c*a)*a-1 ⇒ b=c.
In a non-Abelian group, you must cancel from the same side. You cannot move elements across an equation as if the operation were ordinary commutative multiplication.
11. Abelian Group
A group (G,*) is called an Abelian group (or commutative group) if a*b=b*a for every a,b∈G.
| Structure | Group? | Abelian? | Main reason |
|---|---|---|---|
| (ℤ,+) | Yes | Yes | Integer addition satisfies all group axioms and is commutative. |
| (ℚ,+) | Yes | Yes | Addition is closed, associative, has 0 and additive inverses. |
| ({1,-1},×) | Yes | Yes | Closed under multiplication; identity 1; each element has inverse. |
| (ℤ,−) | No | — | Subtraction is not associative and has no two-sided identity on ℤ. |
| (ℤ,×) | No | — | Most integers do not have multiplicative inverses in ℤ. |
12. Operation Tables / Cayley Tables
For a finite set, a binary operation can be displayed by a table. The entry in row a and column b is a*b.
12.1 Example: G={1,−1} under multiplication
| × | 1 | −1 |
|---|---|---|
| 1 | 1 | −1 |
| −1 | −1 | 1 |
From the table:
- Every entry belongs to G → closure.
- The row and column headed by 1 reproduce the elements → identity is 1.
- 1 is its own inverse and −1 is its own inverse.
- The table is symmetric about the main diagonal → the operation is commutative.
A table makes closure, identity and inverses easy to inspect, and symmetry helps check commutativity. Associativity usually needs a separate argument unless it is inherited from a familiar associative operation.
13. Worked Examples
On ℤ, define a*b=a+b+2. Show that (ℤ,*) is an Abelian group.
- Closure: for integers a,b, a+b+2∈ℤ.
- Associativity: (a*b)*c=(a+b+2)+c+2=a+b+c+4, while a*(b*c)=a+(b+c+2)+2=a+b+c+4.
- Identity: require a*e=a. Thus a+e+2=a, so e=-2.
- Inverse: if b is the inverse of a, then a*b=-2. Hence a+b+2=-2, so b=-a-4, an integer.
- Commutativity: a*b=a+b+2=b+a+2=b*a.
Therefore (ℤ,*) is an Abelian group.
Is (ℤ,−) a group?
Although subtraction is closed on integers, it is not associative:
(5−3)−1=1, but 5−(3−1)=3.
Therefore (ℤ,−) is not a group.
Let G={1,2,3,…} under ordinary addition.
The operation is closed and associative, but the additive identity would have to be 0, and 0∉G. Hence (G,+) is not a group.
Verify that G={1,-1} is a group under multiplication.
- Closure follows from the operation table.
- Multiplication is associative.
- Identity is 1.
- 1-1=1 and (-1)-1=-1.
- Multiplication is commutative.
So this is a finite Abelian group of order 2.
On ℤ, define a*b=a+b+ab. Is ℤ a group under this operation?
The operation is closed and associative. The identity is 0 because a*0=a. For an inverse b:
a+b+ab=0 ⇒ b=−a/(a+1), provided a≠−1.
This is not an integer for every integer a, and for a=-1 no inverse exists. Therefore (ℤ,*) is not a group.
In a group, solve a*x=b for x.
Multiply on the left by a-1:
a-1*(a*x)=a-1*b ⇒ (a-1*a)*x=a-1*b ⇒ x=a-1*b.
The order matters: without commutativity, one cannot write b*a-1 instead.
On ℝ, define a*b=a+b-5. Find the identity.
a*e=a ⇒ a+e−5=a ⇒ e=5.
Also e*a=5+a-5=a. Thus the two-sided identity is 5.
For the operation a*b=a+b-5 on ℝ, identity e=5. Find the inverse of a.
a*b=5 ⇒ a+b−5=5 ⇒ b=10−a.
Hence the inverse of a is 10-a.
14. Quick Comparison Table
| Concept | Condition | Key Question |
|---|---|---|
| Binary operation | *:S×S→S | Does every pair give one result inside S? |
| Associative | (a*b)*c=a*(b*c) | Can brackets move without changing the result? |
| Identity | a*e=e*a=a | Is there a neutral element? |
| Inverse | a*a⁻¹=a⁻¹*a=e | Can every element be returned to the identity? |
| Group | Closure + associativity + identity + inverses | Do all four axioms hold? |
| Finite group | Finitely many elements | What is the order? |
| Infinite group | Infinitely many elements | Does the set continue indefinitely? |
| Abelian group | a*b=b*a | Does order of operands not matter? |
15. Common Mistakes and Warnings
Closure alone does not make a structure a group. All four axioms are required.
For a newly defined operation, prove associativity explicitly unless it clearly follows from a known associative operation.
A group identity must work on both sides: a*e=e*a=a.
An algebraic solution is not enough; the inverse must belong to the group set.
Commutativity is an additional condition and must be checked separately.
In a general group, cancellation must be performed from the same side.
(a*b)-1=b-1*a-1, not generally a-1*b-1.
The identity depends on the operation. It need not be 0 or 1.
16. Exam-Important Results
17. Important Exam Questions
Short-Answer Questions
- Define a binary operation on a non-empty set.
- Give one example and one non-example of a binary operation on ℤ.
- State the four axioms of a group.
- Define the identity element and inverse of an element.
- Define finite group, infinite group and order of a finite group.
- Define an Abelian group.
- Give an example of an infinite Abelian group.
- Explain why (ℤ,×) is not a group.
- State the left and right cancellation laws.
- Find the identity for a custom operation such as a*b=a+b+k.
Proof / Long-Answer Questions
- Prove that the identity element of a group is unique.
- Prove that the inverse of every group element is unique.
- Prove the left cancellation law in a group.
- Prove the right cancellation law in a group.
- Prove that (a-1)-1=a.
- Prove that (a*b)-1=b-1*a-1.
- For a given custom operation, verify all group axioms and determine whether the group is Abelian.
Problem-Solving Questions
- Determine whether a given formula defines a binary operation on ℤ.
- Given a*b=a+b+c, find the identity and the inverse of a general element.
- Test whether a familiar set under addition, subtraction or multiplication is a group.
- Use a finite operation table to identify the identity and inverses.
- Solve equations such as a*x=b or x*a=b in a group.
- Decide whether a verified group is Abelian.
Diagram / Table Questions
- Draw a mapping diagram to explain a binary operation S×S→S.
- Construct and interpret an operation table for a small finite group.
- Draw a flowchart showing the four tests required to verify a group.
18. One-Minute Revision
- A binary operation on S maps every ordered pair in S×S to exactly one element of S.
- Closure means a*b∈S for every a,b∈S.
- Associativity is (a*b)*c=a*(b*c).
- A group requires closure, associativity, an identity and an inverse for every element.
- The identity satisfies a*e=e*a=a.
- The inverse satisfies a*a-1=a-1*a=e.
- The identity element of a group is unique.
- The inverse of each group element is unique.
- Cancellation is valid in groups because inverses exist.
- A finite group has finitely many elements; its number of elements is its order.
- (ℤ,+) is an infinite Abelian group.
- (ℤ,×) is not a group because most elements lack inverses in ℤ.
- An Abelian group is a group whose operation is commutative.
- Do not assume an identity must be 0 or 1; it depends on the operation.
- For custom operations, verify each axiom separately and check that identities and inverses belong to the set.
19. Diagram Practice
Students should be able to redraw and explain these teaching diagrams or tables:
- Binary-operation mapping S×S→S.
- Closure-test flowchart.
- Four group axioms diagram.
- Group-verification flowchart.
- Finite versus infinite group illustration.
- Proof flow for uniqueness of identity.
- Proof flow for uniqueness of inverse.
- Abelian-group subset diagram.
- A simple operation/Cayley table for a finite group.
Also Visit
Original Nepal eNotes page: Class 12 Mathematics Elementary Group Theory Notes.
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