Revision sheet: Abstract Algebra Essentials

Abstract Algebra (Math 113) - Revision Sheet

1. 📌 Essentials

• Algebraic structures: sets with operations satisfying specific axioms (groups, rings, fields).
• Prime number: only divisible by 1 and itself.
• Group: set with an associative binary operation, identity, and inverses.
• Ring: set with addition and multiplication, distributive, with identity.
• Field: commutative ring with multiplicative invers for non-zero elements.
• Congruence mod m: a ≡ b mod m m | (a−b); forms residue classes.
• Cayley's theorem: any group G embeds into a symmetric group Σ(G).
• Normal subgroup: invariant under conjugation; allows quotient groups.
• Simple group: no non-trivial normal subgroups; building block of finite groups.
• Galois extension: normal + separable; automorphisms fixing base field.

2. 🧩 Key Structures & Components

• Set — collection of objects.
• Function — rule from set S to T, with properties: injective, surjective, bijective.
• Equivalence relation — partitions set into classes; reflexive, symmetric, transitive.
• Integers (Z) — prime factorization, gcd, divisibility.
• Residue class — elements of Z/mZ; field iff m prime.
• *Group (G, ) — closure, associativity, identity, inverses.
• Cyclic group — generated by one element; isomorphic to Z or Z/mZ.
• Permutation group Σ(S) — all bijections S→S.
• Normal subgroup (H ◁ G) — stable under conjugation.
• Quotient group G/H — group formed when H normal.
• Ring — set with addition and multiplication, distributive.
• Field — commutative ring with inverses, algebraically closed (C).
• Polynomial ring F[X] — polynomials over field F.
• Galois group — automorphisms fixing base field.

3. 🔬 Functions, Mechanisms & Relationships

• Set functions: compose (f ◦ g), identity Id.
• Equivalence classes: define partitions; relate to quotient sets.
• Prime factorization: unique in Z; fundamental for divisibility.
• Congruences: partition Z into residue classes; structure depends on primality.
• Groups: hierarchy: cyclic ⊆ abelian ⊆ general groups.
• Permutation actions: orbits and stabilizers linked via Orbit-Stabilizer.
• Normal subgroups: allow well-defined quotient groups.
• Isomorphism theorems: relate subgroups, quotients, and automorphisms.
• Galois theory: automorphisms correspond to field extensions; group structure determines solvability.

4. Classification & Summary Table

Item	Key Features	Notes
Set & Function	Sets: collections; functions: rules; composition	Basic language of algebra
Equivalence Relation	Reflexive, symmetric, transitive; partitions	Equivalence classes form partitions
Integers (Z)	Prime, gcd, divisibility, Euclid’s algorithm	Unique prime factorization
Congruence mod m	a ≡ b mod m iff m	(a−b); residue classes Z/mZ
Group (G, *)	Closure, associativity, identity, inverses	Cyclic, abelian, subgroups, cosets
Cyclic Group	Generated by one element; isomorphic to Z or Z/mZ	Fundamental building block
Permutation Group Σ(S)	All bijections; acts on S	Cayley’s theorem: G embeds into Σ(G)
Normal Subgroup	gHg⁻¹ = H; quotient G/H well-defined	Key for constructing quotient groups
Simple Group	No non-trivial normal subgroups	Cyclic prime order, alternating, Lie, sporadic
Ring	Set with +, ×; distributive, identity	Commutative rings, ideals
Field	Commutative ring with inverses; algebraically closed (C)	Basic algebraic structure
Polynomial Ring	Over field F; degree, irreducibility, roots	Factorization, minimal polynomial
Galois Group	Automorphisms fixing base field; order = [E:F]	Determines solvability of polynomials

5. 🗂️ Hierarchical Diagram (ASCII)

Algebraic Structures
 ├─ Sets & Functions
 │    ├─ Equivalence Relations
 │    │    └─ Partitions
 │    └─ Functions (composition, identity)
 ├─ Number Systems
 │    ├─ Integers (Z)
 │    │    ├─ Prime factorization
 │    │    └─ GCD, divisibility
 │    └─ Congruences (mod m)
 │         └─ Residue classes Z/mZ
 ├─ Groups
 │    ├─ Cyclic, abelian, subgroups
 │    ├─ Permutation groups Σ(S)
 │    │    └─ Cayley’s theorem
 │    └─ Normal subgroups & quotient groups
 └─ Rings & Fields
      ├─ Rings: +, ×, ideals
      ├─ Fields: inverses, algebraically closed (C)
      └─ Polynomial rings over F

6. ⚠️ High-Yield Pitfalls & Confusions

• Confusing cyclic with abelian groups.
• Mistaking normal subgroup for any subgroup.
• Assuming all quotient groups are automatically groups (must be normal).
• Believing Z/mZ is always a field (only if m prime).
• Overlooking that automorphisms in Galois groups fix the base field.
• Confusing irreducibility with reducibility over different fields.
• Assuming all polynomials split over any extension (only algebraically closed fields).
• Misunderstanding the difference between separable and normal extensions.

7. ✅ Final Exam Checklist

• Define and identify groups, rings, fields.
• Understand subgroups, normal subgroups, quotients.
• Know Cayley's theorem and its implications.
• Explain congruence mod m and residue class structure.
• State the Fundamental Theorem of Arithmetic.
• Describe cyclic groups and their properties.
• Recognize permutation groups and group actions.
• Understand Galois extensions and Galois groups.
• Apply isomorphism theorems to relate structures.
• Identify simple groups and their classification.
• Use Eisenstein’s Criterion for irreducibility.
• Know the properties of algebraically closed fields.
• Connect Galois groups with solvability of polynomials.
• Understand field extensions: degree, automorphisms, fixed fields.
• Recognize the importance of normality and separability.
• Be familiar with polynomial roots and their bounds.

Strictly high-yield, exam-focused, structured for rapid review and mastery.