TRB BT Assistant - Groups, Subgroups and Cyclic Groups E-Book Study Material

Groups, Subgroups and Cyclic Groups

1. Introduction to Groups

In mathematics, a group is a fundamental algebraic structure. It consists of a set of elements along with an operation that combines any two elements to form a third element. This operation must satisfy four specific conditions, known as the group axioms. These axioms ensure that the structure behaves in a predictable and consistent way, making it a powerful tool in various fields of mathematics, physics, and computer science.

The concept of a group allows us to abstract the properties of symmetry and transformation. For example, the set of rotations that leave a square unchanged, under the operation of composition, forms a group. Similarly, the set of integers under addition forms a group. Understanding group theory helps us analyze these structures and their relationships.

1.1 Definition of a Group

A group is a set G, equipped with a binary operation *, such that the following four axioms hold:

Closure: For any two elements a and b in G, the result of the operation, a * b, is also an element of G.
Associativity: For any elements a, b, and c in G, the equation (a * b) * c = a * (b * c) holds.
Identity Element: There exists an element e in G, such that for every element a in G, the equation e * a = a * e = a holds. This element e is called the identity element.
Inverse Element: For each element a in G, there exists an element a^-1 in G, such that a * a^-1 = a^-1 * a = e, where e is the identity element. This element a^-1 is called the inverse of a.

1.2 Abelian Groups

A group (G, *) is called an Abelian group (or commutative group) if the binary operation * is commutative. This means that for any two elements a and b in G, the equation a * b = b * a holds.

Example: The set of integers (ℤ) with the operation of addition (+) is an Abelian group.

Closure: a + b is an integer if a and b are integers.
Associativity: (a + b) + c = a + (b + c).
Identity Element: 0, because a + 0 = 0 + a = a.
Inverse Element: For any integer a, its inverse is -a, because a + (-a) = (-a) + a = 0.
Commutativity: a + b = b + a.

The set of integers (ℤ) with the operation of multiplication (×) is NOT a group because it lacks an identity element for some elements (e.g., 2 does not have a multiplicative inverse in ℤ) and it does not satisfy closure for all elements (e.g., 1/2 is not an integer).

1.3 Examples of Groups

Let's explore a few more examples to solidify the understanding of group properties.

Example 1: The set of non-zero rational numbers under multiplication. Let Q* be the set of non-zero rational numbers and * be the operation of multiplication.

Closure: The product of two non-zero rational numbers is a non-zero rational number.
Associativity: Multiplication of rational numbers is associative.
Identity Element: 1, since a × 1 = 1 × a = a for any a ∈ Q*.
Inverse Element: For any non-zero rational number a = p/q, its inverse is a^-1 = q/p, since (p/q) × (q/p) = 1.

This forms an Abelian group because a × b = b × a.

Example 2: The set of 2x2 invertible matrices with real entries under matrix multiplication. Let GL(2, ℝ) be the set of 2x2 matrices with real entries and determinant not equal to zero. The operation is matrix multiplication.

Closure: The product of two invertible 2x2 matrices is an invertible 2x2 matrix.
Associativity: Matrix multiplication is associative.
Identity Element: The identity matrix I = [[1, 0], [0, 1]].
Inverse Element: Every invertible matrix has an inverse matrix.

This group is not Abelian, as matrix multiplication is generally not commutative.

Example 3: The set {1, -1, i, -i} under multiplication. Let G = {1, -1, i, -i}.

Closure: Check all possible products. For example, i × i = -1, i × (-i) = 1. All products are in G.
Associativity: Complex number multiplication is associative.
Identity Element: 1.
Inverse Element: 1^-1=1, (-1)^-1=-1, i^-1=-i, (-i)^-1=i.

This is an Abelian group.

1.4 Properties of Groups

Several important properties can be derived from the group axioms. These properties simplify proofs and help in understanding the structure of groups.

Uniqueness of the Identity Element: There is only one identity element in a group.
Uniqueness of Inverses: Each element in a group has a unique inverse.
Cancellation Laws: For any elements a, b, and c in a group G, if a * b = a * c, then b = c (left cancellation). If b * a = c * a, then b = c (right cancellation).
Inverse of a Product: For any two elements a and b in a group G, the inverse of their product is given by (a * b)^-1 = b^-1 * a^-1.

Proof of (a * b)^-1 = b^-1 * a^-1: To show that b^-1 * a^-1 is the inverse of a * b, we need to show that (a * b) * (b^-1 * a^-1) = e and (b^-1 * a^-1) * (a * b) = e. Using associativity and the definition of inverses: (a * b) * (b^-1 * a^-1) = a * (b * b^-1) * a^-1 = a * e * a^-1 = a * a^-1 = e. Similarly, (b^-1 * a^-1) * (a * b) = b^-1 * (a^-1 * a) * b = b^-1 * e * b = b^-1 * b = e. Thus, (a * b)^-1 = b^-1 * a^-1.

2. Subgroups

A subgroup is a subset of a group that is itself a group under the same operation as the parent group. This concept allows us to identify smaller, self-contained group structures within larger ones. Subgroups play a crucial role in understanding the overall structure of a group.

2.1 Definition of a Subgroup

Let (G, *) be a group and H be a non-empty subset of G. H is a subgroup of G if H is closed under the operation * and for every element in H, its inverse is also in H. In other words, H forms a group under the operation *.

Every group G has at least two subgroups: the group G itself and the trivial subgroup {e}, which contains only the identity element.

2.2 Subgroup Test

To verify if a subset H of a group G is a subgroup, we can use the following tests. These tests simplify the verification process by reducing the number of axioms to check.

2.2.1 One-Step Subgroup Test

A non-empty subset H of a group G is a subgroup of G if and only if for any two elements a and b in H, the element a * b^-1 is also in H.

Explanation:

If H is a subgroup, then it is closed under the operation and contains inverses. So, if a ∈ H and b ∈ H, then b^-1 ∈ H, and by closure, a * b^-1 ∈ H.
Conversely, assume a * b^-1 ∈ H for all a, b ∈ H.
- To show closure: Take any a, b ∈ H. Then a * b = a * (b^-1)^-1. Since b ∈ H, b^-1 ∈ H. Then a * (b^-1)^-1 ∈ H. Thus, H is closed.
- To show the existence of an identity element: Take any a ∈ H. Then a * a^-1 = e. So, e ∈ H.
- To show the existence of inverses: Take any a ∈ H. We know e ∈ H. Using the test, e * a^-1 ∈ H. This means a^-1 ∈ H.
Since H is non-empty, contains the identity, is closed, and contains inverses, H is a subgroup.

2.2.2 Two-Step Subgroup Test

A non-empty subset H of a group G is a subgroup of G if and only if:

H is closed under the operation *.
For every element a in H, its inverse a^-1 is also in H.

This is essentially restating the definition of a subgroup.

2.2.3 Finite Subgroup Test

If G is a group and H is a non-empty finite subset of G, then H is a subgroup of G if and only if H is closed under the operation *.

Explanation: For finite sets, closure automatically implies the existence of inverses. If H is finite and closed, consider the sequence a, a², a³, ... for any a ∈ H. Since H is finite, these elements must repeat. Let a^m = aⁿ for m > n ≥ 1. Then a^m-n = e. If m-n > 0, then a^m-n-1 * a = e, so a^m-n-1 is the inverse of a. This inverse is also in H due to closure.

2.3 Examples of Subgroups

Let's examine some concrete examples of subgroups.

Example 1: Integers as a subgroup of Rational Numbers. The set of integers (ℤ, +) is a subgroup of the set of rational numbers (ℚ, +).

H = ℤ, G = ℚ, operation is +.
Check closure: If a, b ∈ ℤ, then a + b ∈ ℤ. (Satisfied)
Check inverses: If a ∈ ℤ, then -a ∈ ℤ. (Satisfied)
Using the one-step test: If a, b ∈ ℤ, then a + (-b) = a - b. Since a and b are integers, a - b is also an integer. So, ℤ is a subgroup of ℚ under addition.

Example 2: Even integers as a subgroup of Integers. The set of even integers (2ℤ, +) is a subgroup of the set of integers (ℤ, +).

H = {..., -4, -2, 0, 2, 4, ...}, G = ℤ, operation is +.
Closure: If a = 2k and b = 2m are even, then a + b = 2k + 2m = 2(k + m), which is also even.
Inverses: If a = 2k is even, then -a = -2k = 2(-k), which is also even.
Thus, 2ℤ is a subgroup of ℤ under addition.

Example 3: Multiplicative Group of Integers Modulo n. Consider the group (ℤ_n*, ×), where ℤ_n* is the set of integers between 1 and n-1 that are relatively prime to n, and × denotes multiplication modulo n. Let's take n=5. ℤ₅* = {1, 2, 3, 4}. The operation is multiplication modulo 5.

Identity: 1.
Closure: Check all products. e.g., 2 × 3 = 6 ≡ 1 (mod 5), 3 × 4 = 12 ≡ 2 (mod 5). All products are in {1, 2, 3, 4}.
Inverses: 1^-1=1, 2^-1=3 (since 2×3=6≡1), 3^-1=2, 4^-1=4 (since 4×4=16≡1). All inverses are in the set.
Associativity: Multiplication modulo n is associative.

So, (ℤ₅*, ×) is a group. If we consider a subset, for instance, {1, 4}.

1 × 1 = 1, 1 × 4 = 4, 4 × 1 = 4, 4 × 4 = 16 ≡ 1 (mod 5).
Closure holds. Inverses: 1^-1=1, 4^-1=4. Both exist within the subset.

Thus, {1, 4} is a subgroup of (ℤ₅*, ×).

3. Cyclic Groups

A cyclic group is a group that can be generated by a single element. This means that every element in the group can be expressed as a power of this single generator. Cyclic groups are the simplest type of groups and are fundamental to understanding group theory.

3.1 Definition of a Cyclic Group

A group G is called cyclic if there exists an element g in G such that every element of G can be written as an integer power of g. The element g is called a generator of G. If G is cyclic with generator g, then G = {gⁿ | n ∈ ℤ}.

3.2 Finite Cyclic Groups

A finite cyclic group of order n is isomorphic to the group (ℤ_n, +) of integers modulo n under addition.

Let G be a finite cyclic group of order n with generator g. Then G = {g⁰, g¹, g², ..., g^n-1}. The operation is multiplication: gⁱ * g^j = g^{(i+j) mod n}. The identity element is g⁰ = e. The inverse of g^k is g^n-k (mod n).

Example: Consider the group ℤ₆ = {0, 1, 2, 3, 4, 5} under addition modulo 6. The order of this group is 6. Let's check if 1 is a generator:

1¹ = 1
1² = 1 + 1 = 2
1³ = 1 + 1 + 1 = 3
1⁴ = 4
1⁵ = 5
1⁶ = 6 ≡ 0 (mod 6) (This is the identity element, so the order of the element is 6)

Since 1 generates all elements of ℤ₆, 1 is a generator. Thus, ℤ₆ is a cyclic group.

Let's check if 2 is a generator:

2¹ = 2
2² = 2 + 2 = 4
2³ = 2 + 2 + 2 = 6 ≡ 0 (mod 6)

The powers of 2 only produce {0, 2, 4}. The order of element 2 is 3. So, 2 is not a generator of ℤ₆. The subgroup generated by 2 is {0, 2, 4}, which is a cyclic subgroup of order 3.

Let's check if 3 is a generator:

3¹ = 3
3² = 3 + 3 = 6 ≡ 0 (mod 6)

The powers of 3 produce {0, 3}. The order of element 3 is 2. The subgroup generated by 3 is {0, 3}, a cyclic subgroup of order 2.

Let's check if 5 is a generator:

5¹ = 5
5² = 5 + 5 = 10 ≡ 4 (mod 6)
5³ = 5 + 5 + 5 = 15 ≡ 3 (mod 6)
5⁴ = 4 + 5 = 9 ≡ 3 (mod 6) - mistake here, powers are cumulative.
5⁴ = 5 + 5 + 5 + 5 = 20 ≡ 2 (mod 6)
5⁵ = 5 + 5 + 5 + 5 + 5 = 25 ≡ 1 (mod 6)
5⁶ = 5 + 5 + 5 + 5 + 5 + 5 = 30 ≡ 0 (mod 6)

The powers of 5 generate {0, 1, 2, 3, 4, 5}. So, 5 is a generator. The order of element 5 is 6.

Shortcut for Generators of ℤ_n: An element k in ℤ_n is a generator if and only if gcd(k, n) = 1. For ℤ₆: gcd(1, 6) = 1 (Generator) gcd(2, 6) = 2 (Not a generator) gcd(3, 6) = 3 (Not a generator) gcd(4, 6) = 2 (Not a generator) gcd(5, 6) = 1 (Generator) The generators are 1 and 5.

3.3 Infinite Cyclic Groups

An infinite cyclic group is isomorphic to the group of integers (ℤ, +) under addition.

Let G be an infinite cyclic group with generator g. Then G = {..., g^-2, g^-1, g⁰, g¹, g², ...}. The operation is multiplication: gⁱ * g^j = g^i+j. The identity element is g⁰ = e. The inverse of g^k is g^-k.

The group (ℤ, +) is an infinite cyclic group with generators 1 and -1.

Powers of 1: ..., 1^-2=-2, 1^-1=-1, 1⁰=0, 1¹=1, 1²=2, ... which is all of ℤ.
Powers of -1: ..., (-1)^-2=1, (-1)^-1=-1, (-1)⁰=0, (-1)¹=-1, (-1)²=1, ... mistake here, powers are additive in ℤ.

Correcting for (ℤ, +): Let g = 1.

1 * n = n (repeated addition of 1, n times).
1 * 0 = 0 (identity)
1 * (-n) = -n (repeated addition of 1, -n times, which is subtraction of 1, n times)

So, the generator 1 produces all integers. Let g = -1.

(-1) * n = -n.
(-1) * 0 = 0
(-1) * (-n) = n

So, the generator -1 also produces all integers.

3.4 Properties of Cyclic Groups

Cyclic groups have several important properties:

Every cyclic group is Abelian.
Every subgroup of a cyclic group is cyclic.
A finite cyclic group of order n has exactly one subgroup of order d for every divisor d of n.
The order of an element in a cyclic group is equal to the order of the subgroup generated by that element.
If G is a cyclic group of order n, then for any two elements a, b ∈ G, the order of a is equal to the order of b if and only if a and b generate the same subgroup.

To find the order of an element 'a' in a finite cyclic group G of order n: The order of 'a' is n / gcd(k, n), where 'a' is represented as g^k and g is a generator of G. Example: In ℤ₁₂ (which is cyclic, generated by 1), find the order of the element 8. Here, n = 12. The element 8 can be written as 1⁸, so k = 8. gcd(8, 12) = 4. Order of 8 = 12 / gcd(8, 12) = 12 / 4 = 3. Let's verify: 8¹ = 8 8² = 8 + 8 = 16 ≡ 4 (mod 12) 8³ = 8 + 8 + 8 = 24 ≡ 0 (mod 12) (Identity element) The order is indeed 3.

3.5 Cyclic Subgroups

As mentioned, every element in a group generates a cyclic subgroup. The subgroup generated by an element 'a' is denoted by ⟨a⟩. ⟨a⟩ = {aⁿ | n ∈ ℤ}

If G is a finite group of order m, then the order of the subgroup ⟨a⟩ must divide m (Lagrange's Theorem). The order of the element 'a' is equal to the order of the cyclic subgroup ⟨a⟩.

Example: In the group of integers modulo 10 under addition (ℤ₁₀, +). Let's find the cyclic subgroup generated by 4, ⟨4⟩.

4¹ = 4
4² = 4 + 4 = 8
4³ = 4 + 4 + 4 = 12 ≡ 2 (mod 10)
4⁴ = 4 + 4 + 4 + 4 = 16 ≡ 6 (mod 10)
4⁵ = 6 + 4 = 10 ≡ 0 (mod 10) (Identity element)

So, ⟨4⟩ = {0, 2, 4, 6, 8}. The order of this subgroup is 5. The order of the element 4 is also 5. This subgroup ⟨4⟩ is cyclic, with generators 4 and 2 (since 2¹=2, 2²=4, 2³=6, 2⁴=8, 2⁵=10≡0).