🌟 Level 3 - Topic 18: Introduction to Group Theory (Part 2) - Structure & Key Theorems 🗝️🏆

1) Homomorphisms and Isomorphisms - Mapping Between Groups 🗺️

In mathematics, when we study structures, it's essential to understand how these structures relate to each other. Homomorphisms and Isomorphisms are mappings between groups that preserve the group operation. They are crucial for understanding the relationships and similarities between different groups.

Let \( (G, *) \) and \( (G', \diamond) \) be two groups.

A Homomorphism from \(G\) to \(G'\) is a function \( \phi: G \to G' \) such that for all \( a, b \in G \), \( \phi(a * b) = \phi(a) \diamond \phi(b) \). (Operation-preserving map)
An Isomorphism from \(G\) to \(G'\) is a homomorphism \( \phi: G \to G' \) that is also bijective (both injective and surjective). If there exists an isomorphism between \(G\) and \(G'\), we say \(G\) and \(G'\) are isomorphic, denoted \( G \cong G' \). (Structure-preserving and bijective map - essentially "same" group structure)

Example 1: Homomorphism - from \( (\mathbb{Z}, +) \) to \( (\mathbb{Z}_n, +_n) \)

Consider the map \( \phi: (\mathbb{Z}, +) \to (\mathbb{Z}_n, +_n) \) defined by \( \phi(a) = a \pmod{n} \) (reduction modulo \(n\)). Is this a homomorphism?

Yes. We need to check if \( \phi(a + b) = \phi(a) +_n \phi(b) \) for all \( a, b \in \mathbb{Z} \).

\( \phi(a + b) = (a + b) \pmod{n} \) and \( \phi(a) +_n \phi(b) = (a \pmod{n}) +_n (b \pmod{n}) = (a \pmod{n} + b \pmod{n}) \pmod{n} = (a + b) \pmod{n} \).

So, \( \phi(a + b) = \phi(a) +_n \phi(b) \). Thus, \( \phi \) is a homomorphism. It is also surjective (onto), but not injective (not one-to-one), so it is not an isomorphism. It's called a natural homomorphism or canonical homomorphism.

Example 2: Isomorphism - \( (\mathbb{Z}, +) \) and \( (2\mathbb{Z}, +) \) ? No.

Are \( (\mathbb{Z}, +) \) and \( (2\mathbb{Z}, +) \) isomorphic? (Where \( 2\mathbb{Z} \) is the subgroup of even integers).

Let's consider if there could be an isomorphism \( \phi: (\mathbb{Z}, +) \to (2\mathbb{Z}, +) \). If it were, it would need to be bijective and operation-preserving. Suppose such an isomorphism existed. Consider the generator \( 1 \) of \( (\mathbb{Z}, +) \). If \( \phi \) is an isomorphism, then \( \phi(1) \) must generate \( (2\mathbb{Z}, +) \). Let \( \phi(1) = 2k \) for some integer \(k\) (since \( \phi(1) \in 2\mathbb{Z} \)). Then for any integer \( n \), \( \phi(n) = \phi(1 + 1 + ... + 1) = \phi(1) + \phi(1) + ... + \phi(1) = n \cdot \phi(1) = n \cdot (2k) = 2nk \). So, the image of \( \phi \) is \( \{ 2nk \mid n \in \mathbb{Z} \} = 2k\mathbb{Z} \). If \( k = \pm 1 \), then image is \( 2\mathbb{Z} \). If \( k = 0 \), image is \( \{0\} \). For any \( k \neq \pm 1, 0 \), the image is a proper subgroup of \( 2\mathbb{Z} \), not \( 2\mathbb{Z} \) itself. So, to be surjective onto \( 2\mathbb{Z} \), we need \( k = \pm 1 \). Let's try \( \phi(n) = 2n \). Then \( \phi: (\mathbb{Z}, +) \to (2\mathbb{Z}, +) \) is defined by multiplying by 2.

Homomorphism?: \( \phi(m + n) = 2(m + n) = 2m + 2n = \phi(m) + \phi(n) \). Yes, it's a homomorphism.
Injective?: If \( \phi(m) = \phi(n) \), then \( 2m = 2n \), so \( m = n \). Yes, injective (one-to-one).
Surjective?: Is every element in \( 2\mathbb{Z} \) in the image of \( \phi \)? Let \( y \in 2\mathbb{Z} \). Then \( y = 2k \) for some integer \( k \). Is there \( x \in \mathbb{Z} \) such that \( \phi(x) = y \)? Yes, take \( x = k \). Then \( \phi(k) = 2k = y \). Yes, surjective (onto).

Thus, \( \phi(n) = 2n \) is an isomorphism from \( (\mathbb{Z}, +) \) to \( (2\mathbb{Z}, +) \). So, \( (\mathbb{Z}, +) \cong (2\mathbb{Z}, +) \). This might be surprising - \( 2\mathbb{Z} \) is a proper *subset* of \( \mathbb{Z} \), but they have the "same group structure"! This is possible for infinite groups. For finite groups, a proper subgroup can never be isomorphic to the whole group.

Example 3: Kernel of a Homomorphism

For the homomorphism \( \phi: (\mathbb{Z}, +) \to (\mathbb{Z}_n, +_n) \) defined by \( \phi(a) = a \pmod{n} \), what is the kernel?

The Kernel of a homomorphism \( \phi: G \to G' \) is the set of elements in \(G\) that are mapped to the identity element \(e'\) of \(G'\). Denoted \( \ker(\phi) = \{ g \in G \mid \phi(g) = e' \} \). The kernel is always a normal subgroup of \(G\) (we'll define normal subgroup later).

In our example, the identity element in \( (\mathbb{Z}_n, +_n) \) is \( 0 \). So, \( \ker(\phi) = \{ a \in \mathbb{Z} \mid \phi(a) = 0 \pmod{n} \} = \{ a \in \mathbb{Z} \mid a \equiv 0 \pmod{n} \} = \{ a \in \mathbb{Z} \mid a = kn \text{ for some integer } k \} = n\mathbb{Z} \). Thus, the kernel of \( \phi \) is \( n\mathbb{Z} \), the set of integer multiples of \(n\). For example, if \( n = 4 \), \( \ker(\phi) = 4\mathbb{Z} = \{ ..., -8, -4, 0, 4, 8, ... \} \).

2) Lagrange's Theorem - Orders and Subgroups 📜

Lagrange's Theorem is a fundamental result in finite group theory. It establishes a strong constraint on the possible sizes (orders) of subgroups of a finite group.

Lagrange's Theorem

Theorem: Let \(G\) be a finite group and \(H\) be a subgroup of \(G\). Then the order of \(H\) (number of elements in \(H\), denoted \(|H|\)) divides the order of \(G\) (number of elements in \(G\), denoted \(|G|\)). In other words, \( |H| \) is a divisor of \( |G| \).

Furthermore, the index of \(H\) in \(G\), defined as the number of distinct left (or right) cosets of \(H\) in \(G\), is given by \( [G:H] = \frac{|G|}{|H|} \), which is an integer.

Example 4: Subgroups of \( \mathbb{Z}_{12} \) and Lagrange's Theorem

Consider the cyclic group \( (\mathbb{Z}_{12}, +_{12}) \) of order \( 12 \). By Lagrange's Theorem, the order of any subgroup of \( \mathbb{Z}_{12} \) must divide 12. Divisors of 12 are 1, 2, 3, 4, 6, 12. So, possible subgroup orders are 1, 2, 3, 4, 6, 12. And indeed, for each divisor \(d\) of \(12\), there is exactly one subgroup of order \(d\) in \( \mathbb{Z}_{12} \). For example:

Order 1: \( \{ 0 \} \) (trivial subgroup)
Order 2: \( \{ 0, 6 \} \)
Order 3: \( \{ 0, 4, 8 \} \)
Order 4: \( \{ 0, 3, 6, 9 \} \)
Order 6: \( \{ 0, 2, 4, 6, 8, 10 \} \)
Order 12: \( \mathbb{Z}_{12} \) itself (the whole group)

Lagrange's Theorem is very powerful - it restricts the possible subgroup structures of a finite group dramatically. It's a cornerstone result in finite group theory.

3) Normal Subgroups and Quotient Groups - Factor Groups ➗

In group theory, sometimes we can "divide" one group by a special type of subgroup, called a normal subgroup, to create a new group called a quotient group or factor group. This construction is analogous to factoring out subspaces in linear algebra or ideals in ring theory.

A subgroup \(N\) of a group \(G\) is called a Normal Subgroup of \(G\), denoted \( N \trianglelefteq G \), if for all \( g \in G \) and \( n \in N \), \( gng^{-1} \in N \). (Conjugation of any element of \(N\) by any element of \(G\) remains in \(N\)). In Abelian groups, every subgroup is normal.

If \(N\) is a normal subgroup of \(G\), we can form the Quotient Group (or Factor Group) \( G/N \). The elements of \( G/N \) are the cosets of \(N\) in \(G\). The group operation in \( G/N \) is defined by \( (aN) * (bN) = (a*b)N \). The order of \( G/N \) is \( |G/N| = \frac{|G|}{|N|} = [G:N] \).

Example 5: Quotient Group \( \mathbb{Z} / n\mathbb{Z} \cong \mathbb{Z}_n \)

Consider the normal subgroup \( n\mathbb{Z} \) of \( (\mathbb{Z}, +) \). (Every subgroup of Abelian group is normal). The quotient group \( \mathbb{Z} / n\mathbb{Z} \) consists of cosets of \( n\mathbb{Z} \) in \( \mathbb{Z} \). These cosets are of the form \( a + n\mathbb{Z} = \{ a + nk \mid k \in \mathbb{Z} \} \). There are exactly \(n\) distinct cosets:

\( 0 + n\mathbb{Z}, 1 + n\mathbb{Z}, 2 + n\mathbb{Z}, ..., (n-1) + n\mathbb{Z} \)

The operation in \( \mathbb{Z} / n\mathbb{Z} \) is coset addition: \( (a + n\mathbb{Z}) + (b + n\mathbb{Z}) = (a + b) + n\mathbb{Z} \). This is exactly how addition modulo \(n\) works! In fact, \( \mathbb{Z} / n\mathbb{Z} \) is isomorphic to \( \mathbb{Z}_n \). We have \( \mathbb{Z} / n\mathbb{Z} \cong \mathbb{Z}_n \).

Example 6: Quotient Group - Example with Dihedral Group (Non-Abelian, more advanced - optional)

(Example with Dihedral Group \(D_4\) and a normal subgroup, showing construction of quotient group - may be omitted or briefly mentioned as more advanced example, depending on desired level of depth).

4) Direct Products of Groups - Building Larger Groups 🏗️

Given two groups, we can construct a new, larger group called their direct product. This is a way to combine group structures.

Let \( (G, *) \) and \( (H, \diamond) \) be two groups. The External Direct Product of \(G\) and \(H\), denoted \( G \times H \), is the group whose underlying set is the Cartesian product \( G \times H = \{ (g, h) \mid g \in G, h \in H \} \). The group operation in \( G \times H \) is defined component-wise:

\( (g_1, h_1) \cdot (g_2, h_2) = (g_1 * g_2, h_1 \diamond h_2) \) for \( (g_1, h_1), (g_2, h_2) \in G \times H \).

Example 7: Direct Product \( \mathbb{Z}_2 \times \mathbb{Z}_3 \)

Consider \( \mathbb{Z}_2 = \{ 0, 1 \} \) and \( \mathbb{Z}_3 = \{ 0, 1, 2 \} \) under addition modulo 2 and 3 respectively. The direct product \( \mathbb{Z}_2 \times \mathbb{Z}_3 \) is the set of ordered pairs \( \{ (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2) \} \). The operation is component-wise addition (first component mod 2, second mod 3). For example:

\( (1, 2) + (1, 1) = (1 +_2 1, 2 +_3 1) = (0, 0) \) (Identity element in \( \mathbb{Z}_2 \times \mathbb{Z}_3 \) is \( (0, 0) \)).

It turns out that \( \mathbb{Z}_2 \times \mathbb{Z}_3 \) is isomorphic to \( \mathbb{Z}_6 \). In general, \( \mathbb{Z}_m \times \mathbb{Z}_n \cong \mathbb{Z}_{mn} \) if and only if \( \gcd(m, n) = 1 \) (m and n are relatively prime).

5) Fundamental Theorem of Finitely Generated Abelian Groups - Structure Theorem (Brief Overview) 🏆

One of the major achievements in group theory is understanding the structure of certain classes of groups. For finitely generated Abelian groups, we have a powerful classification theorem:

Fundamental Theorem of Finitely Generated Abelian Groups (Simplified Statement)

Theorem: Every finitely generated Abelian group \(G\) is isomorphic to a direct product of cyclic groups of the form:

\( \mathbb{Z}_{p_1^{k_1}} \times \mathbb{Z}_{p_2^{k_2}} \times ... \times \mathbb{Z}_{p_r^{k_r}} \times \mathbb{Z} \times \mathbb{Z} \times ... \times \mathbb{Z} \)

where \(p_1, p_2, ..., p_r\) are prime numbers (not necessarily distinct), and \(k_1, k_2, ..., k_r\) are positive integers. The decomposition is unique up to isomorphism and reordering of factors. The number of copies of \( \mathbb{Z} \) in the direct product is called the rank of the group.

This theorem tells us that any finitely generated Abelian group can be built up from basic cyclic groups \( \mathbb{Z}_{p^k} \) and \( \mathbb{Z} \) using direct products. It provides a complete classification of such groups. The proof is more advanced and usually covered in higher-level abstract algebra courses.

Example 8: Applying the Fundamental Theorem

Consider an Abelian group of order 12. By Fundamental Theorem, possible structures are (up to isomorphism):

\( \mathbb{Z}_{12} \) (cyclic group of order 12)
\( \mathbb{Z}_2 \times \mathbb{Z}_6 \) (since \( 2 \cdot 6 = 12 \). And \( \mathbb{Z}_6 \cong \mathbb{Z}_2 \times \mathbb{Z}_3 \), so \( \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \cong \mathbb{Z}_2 \times \mathbb{Z}_6 \))
\( \mathbb{Z}_3 \times \mathbb{Z}_4 \) (since \( 3 \cdot 4 = 12 \). And \( \mathbb{Z}_4 = \mathbb{Z}_{2^2} \), \( \mathbb{Z}_3 = \mathbb{Z}_{3^1} \). This is in the form of the theorem.)
\( \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \) ( \( 2 \cdot 2 \cdot 3 = 12 \). All prime power cyclic groups.)

Actually, \( \mathbb{Z}_2 \times \mathbb{Z}_6 \cong \mathbb{Z}_2 \times (\mathbb{Z}_2 \times \mathbb{Z}_3) = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \). And \( \mathbb{Z}_3 \times \mathbb{Z}_4 \cong \mathbb{Z}_3 \times \mathbb{Z}_{2^2} \). The distinct structures are \( \mathbb{Z}_{12} \), \( \mathbb{Z}_2 \times \mathbb{Z}_6 \cong \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \), and \( \mathbb{Z}_3 \times \mathbb{Z}_4 \). However, \( \mathbb{Z}_{12} \cong \mathbb{Z}_3 \times \mathbb{Z}_4 \) since \( \gcd(3, 4) = 1 \). So really, we have only two distinct isomorphism types for Abelian groups of order 12: \( \mathbb{Z}_{12} \cong \mathbb{Z}_3 \times \mathbb{Z}_4 \) and \( \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \cong \mathbb{Z}_2 \times \mathbb{Z}_6 \). Corrected distinct types of Abelian groups of order 12 are: \( \mathbb{Z}_{12} \) and \( \mathbb{Z}_2 \times \mathbb{Z}_6 \). Or in terms of prime power decomposition: \( \mathbb{Z}_{2^2} \times \mathbb{Z}_3 \) and \( \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \).

6) Brief Glimpse of Applications of Group Theory ✨

Group Theory is not just an abstract mathematical structure; it has profound applications across various fields:

Symmetry and Geometry: Groups describe symmetries of geometric objects (rotations, reflections). Study of symmetry groups (e.g., symmetry groups of molecules in chemistry, crystal groups in physics).
Physics: Symmetry principles in physics are deeply rooted in group theory. Particle physics, quantum mechanics, and crystallography heavily use group representations and symmetry groups.
Cryptography and Coding Theory: Groups are used in constructing cryptographic systems and error-correcting codes. Elliptic curve cryptography, algebraic coding theory.
Computer Science: Abstract data types, algorithms, and complexity theory can be analyzed using group-theoretic tools. Permutation groups are relevant in sorting and searching algorithms.
Art and Music: Symmetries in art patterns (wallpaper groups, frieze groups). Mathematical structures in music theory (group theory can model musical transformations).

This is just a glimpse, but it highlights the wide-ranging impact of group theory beyond pure mathematics.

7) Practice Questions 🎯

7.1 Fundamental – Build Skills

1. Determine if the map \( \phi: (\mathbb{Z}, +) \to (\mathbb{Z}, +) \) defined by \( \phi(x) = 3x \) is a group homomorphism. Is it an isomorphism?

2. Let \( G = (\mathbb{R} \setminus \{0\}, \cdot) \) and \( G' = (\mathbb{R} \setminus \{0\}, \cdot) \). Is the map \( \psi: G \to G' \) defined by \( \psi(x) = x^2 \) a group homomorphism? Why or why not?

3. For the homomorphism \( \phi: (\mathbb{Z}, +) \to (\mathbb{Z}_4, +_4) \) defined by \( \phi(a) = a \pmod{4} \), find the kernel \( \ker(\phi) \).

4. List all possible orders of subgroups of a group of order 15 (using Lagrange's Theorem).

5. List all possible orders of subgroups of a group of order 17 (using Lagrange's Theorem).

6. Is every subgroup of an Abelian group necessarily normal? Explain.

7. Consider the direct product group \( \mathbb{Z}_2 \times \mathbb{Z}_2 \). List all elements of this group and write out its addition table.

8. Is \( \mathbb{Z}_2 \times \mathbb{Z}_2 \) isomorphic to \( \mathbb{Z}_4 \)? Justify your answer by comparing their properties (e.g., are they cyclic?).

9. Using the Fundamental Theorem of Finitely Generated Abelian Groups, list the distinct isomorphism types of Abelian groups of order 8.

10. Give a real-world example (outside of pure mathematics) where group theory is applied (based on section 6 of this topic).

7.2 Challenging – Push Limits 💪🚀

1. Let \( \phi: G \to G' \) be a group homomorphism. Prove that \( \ker(\phi) \) is a subgroup of \( G \). (You can use the One-Step Subgroup Test).

2. Let \( \phi: G \to G' \) be a group homomorphism, and let \( e' \) be the identity in \( G' \). Prove that \( \phi(e) = e' \), where \( e \) is the identity in \( G \). (Homomorphisms map identity to identity).

3. Prove that if \( \phi: G \to G' \) is a group homomorphism, then for any \( a \in G \), \( \phi(a^{-1}) = (\phi(a))^{-1} \). (Homomorphisms preserve inverses).

4. Explain why there is no simple way to "classify" all groups in general (not just Abelian groups like in the Fundamental Theorem). Group theory is very rich and diverse.

5. (Research/Exploration) Look up "First Isomorphism Theorem for Groups". State the theorem and briefly explain in your own words what it means and why it is important. How does it relate kernels, images, and quotient groups?

8) Summary 🎉

Homomorphisms: Operation-preserving maps between groups. Isomorphisms: Bijective homomorphisms (structure-preserving equivalences).
Kernel of Homomorphism: Set of elements mapped to identity, always a normal subgroup.
Lagrange's Theorem: Order of a subgroup divides the order of the finite group. Strong restriction on subgroup sizes.
Normal Subgroups: Special subgroups allowing construction of quotient groups.
Quotient Groups (Factor Groups): Formed by "dividing" a group by a normal subgroup. Elements are cosets. Example: \( \mathbb{Z} / n\mathbb{Z} \cong \mathbb{Z}_n \).
Direct Products: Constructing larger groups from smaller ones using component-wise operations. Example: \( \mathbb{Z}_2 \times \mathbb{Z}_3 \cong \mathbb{Z}_6 \).
Fundamental Theorem of Finitely Generated Abelian Groups: Classifies all finitely generated Abelian groups in terms of direct products of cyclic groups. Provides structure understanding.
Applications of Group Theory: Wide range of applications in symmetry, physics, cryptography, computer science, art, etc.

Congratulations on completing Level 3 Group Theory! You've now explored fundamental concepts, key theorems, and glimpsed the power and breadth of group theory. From the abstract axioms to structural theorems and real-world applications, you've gained a solid foundation in this essential area of mathematics. Keep exploring and deepening your understanding – the world of abstract algebra is rich and rewarding! 🏆🗝️