Group Theory Basics: Groups, Subgroups, Cyclic Groups & Homomorphism

TAKEAWAYS:

• A group is a set with a single operation that satisfies closure, associativity, identity, and inverses.
• Subgroups are subsets of a group that form a group under the same operation.
• Cyclic groups are groups generated by a single element (e.g., integers under addition).
• Homomorphisms map groups to groups while preserving structure (e.g., $f(a\cdot b)=f(a)\cdot f(b)$).
• Lagrange’s Theorem states that the order of a subgroup divides the order of the group.
• Applications include cryptography, symmetry in physics, and error-correcting codes.

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1. What is a Group?

A group is a mathematical structure consisting of a set $G$ and a binary operation $\cdot$ (often called multiplication) that combines any two elements $a,b\in G$ to form another element $a\cdot b\in G$. For $G$ to be a group, it must satisfy four axioms:

Group Axioms (Definition)

Let $(G,\cdot )$ be a set with a binary operation $\cdot$. Then $G$ is a group if:

1. Closure: For all $a,b\in G$, $a\cdot b\in G$.
2. Associativity: For all $a,b,c\in G$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
3. Identity Element: There exists an element $e\in G$ such that for all $a\in G$, $e\cdot a=a\cdot e=a$.
4. Inverse Element: For each $a\in G$, there exists an element $a^{-1}\in G$ such that $a\cdot a^{-1}=a^{-1}\cdot a=e$.

Examples of Groups

Worked Example 1: Verify if $({\mathbb{Z}}_4,+)$ is a Group

Given: ${\mathbb{Z}}_4=\{0,1,2,3\}$ with addition modulo 4. Check axioms:

1. Closure: $1+3=0$ (since $4\equiv 0\mathrm{mod}4$), so closed.
2. Associativity: Inherited from $\mathbb{Z}$.
3. Identity: $0$ is the identity since $a+0=a$.
4. Inverses:
   • $0^{-1}=0$
   • $1^{-1}=3$ (since $1+3=0$)
   • $2^{-1}=2$ (since $2+2=4\equiv 0\mathrm{mod}4$)
   • $3^{-1}=1$

Conclusion: $({\mathbb{Z}}_4,+)$ is a group.

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2. Subgroups

A subgroup $H$ of a group $G$ is a subset of $G$ that is itself a group under the operation of $G$.

Subgroup Test (Lagrange’s Theorem)

A non-empty subset $H$ of $G$ is a subgroup if for all $a,b\in H$, $a\cdot b^{-1}\in H$.

Worked Example 2: Find Subgroups of ${\mathbb{Z}}_6$

Given: ${\mathbb{Z}}_6=\{0,1,2,3,4,5\}$ under addition modulo 6. Possible subgroups:

1. $\{0\}$ (trivial subgroup).
2. $\{0,3\}$:
   • Closure: $3+3=6\equiv 0\mathrm{mod}6$.
   • Identity: $0$.
   • Inverses: $3^{-1}=3$.
3. ${\mathbb{Z}}_6$ itself.

Conclusion: Subgroups are $\{0\}$, $\{0,3\}$, and ${\mathbb{Z}}_6$.

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3. Cyclic Groups

A group $G$ is cyclic if there exists an element $g\in G$ (called a generator) such that every element of $G$ can be written as a power of $g$.

Properties of Cyclic Groups

• Every cyclic group is abelian (commutative).
• If $G$ is cyclic of order $n$, then $G\cong {\mathbb{Z}}_n$.

Worked Example 3: Show ${\mathbb{Z}}_5$ is Cyclic

Given: ${\mathbb{Z}}_5=\{0,1,2,3,4\}$ under addition modulo 5. Check:

• $1$ is a generator because:
  • $1\times 1=1$
  • $1\times 2=2$
  • $1\times 3=3$
  • $1\times 4=4$
  • $1\times 5=5\equiv 0\mathrm{mod}5$.

Conclusion: ${\mathbb{Z}}_5$ is cyclic with generator $1$.

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4. Group Homomorphisms

A homomorphism $f:G\to H$ between two groups is a function that preserves the group operation: $$f(a\cdot b)=f(a)\cdot f(b)\forall a,b\in G.$$

Kernel and Image of a Homomorphism

• Kernel: $\ker (f)=\{g\in G\mid f(g)=e_H\}$ (always a subgroup).
• Image: $\text{Im}(f)=\{f(g)\mid g\in G\}$ (always a subgroup of $H$).

Worked Example 4: Define a Homomorphism $f:\mathbb{Z}\to {\mathbb{Z}}_4$

Given: $f(n)=n\mathrm{mod}4$. Check:

• $f(a+b)=(a+b)\mathrm{mod}4=(a\mathrm{mod}4+b\mathrm{mod}4)\mathrm{mod}4=f(a)+f(b)$.
• Kernel: $\ker (f)=\{n\in \mathbb{Z}\mid n\equiv 0\mathrm{mod}4\}=4\mathbb{Z}$.
• Image: $\text{Im}(f)={\mathbb{Z}}_4$.

Conclusion: $f$ is a homomorphism.

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5. Lagrange’s Theorem

Statement: If $H$ is a subgroup of a finite group $G$, then the order of $H$ divides the order of $G$: $$|G|=|H|\cdot [G:H],$$ where $[G:H]$ is the index of $H$ in $G$.

Worked Example 5: Apply Lagrange’s Theorem to $S_3$

Given: $S_3$ (symmetries of a triangle) has order $6$. Subgroups:

1. $\{e\}$ (order 1).
2. $\{e,(12)\}$ (order 2).
3. $\{e,(123),(132)\}$ (order 3).
4. $S_3$ itself (order 6).

Check: All subgroup orders (1, 2, 3, 6) divide 6.

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6. Applications of Group Theory

1. Cryptography: Used in RSA encryption (modular arithmetic groups).
2. Physics: Symmetry groups in quantum mechanics.
3. Chemistry: Molecular symmetry (point groups).
4. Computer Science: Error-correcting codes (e.g., Reed-Solomon codes).

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Exam Tip

1. Memorize the 4 group axioms and verify them step-by-step in examples.
2. Practice subgroup tests (use the inverse-closure rule).
3. Identify cyclic groups by checking generators.
4. For homomorphisms, always verify $f(a\cdot b)=f(a)\cdot f(b)$.
5. Lagrange’s Theorem is often tested with finite groups—remember subgroup orders must divide the group order.
6. NEB-style questions may ask:
   • "Prove $({\mathbb{Z}}_n,+)$ is a group."
   • "Find all subgroups of ${\mathbb{Z}}_8$."
   • "Define a homomorphism and find its kernel."

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NEB Board-Style Questions

1. Short Answer:

   • Define a group. Give an example of a group and a non-group.
   • What is a cyclic group? Is ${\mathbb{Z}}_6$ cyclic? Justify.

2. Long Answer:

   • Let $G=\{1,-1,i,-i\}$ with multiplication modulo $i^2=-1$. Show $G$ is a group.
   • Find all subgroups of ${\mathbb{Z}}_{12}$. Use Lagrange’s Theorem to verify.

3. Problem Solving:

   • Define a homomorphism $f:\mathbb{Z}\to {\mathbb{Z}}_3$ by $f(n)=n\mathrm{mod}3$. Find $\ker (f)$ and $\text{Im}(f)$.
   • Prove that the set of even integers under addition is a subgroup of $(\mathbb{Z},+)$.

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Summary Table of Key Concepts

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Final Note: Group theory is abstract but powerful! Focus on examples and verification of axioms. Practice with small groups like ${\mathbb{Z}}_n$ and permutation groups to build intuition. Good luck! 🚀