Abstract algebra studies algebraic structures—groups, rings, fields, and modules—defined by sets with operations satisfying axioms. It underlies modern cryptography (RSA, ECC, AES), coding theory (Reed-Solomon, BCH), and computer algebra systems. This cheat sheet covers group theory through Galois theory, domain hierarchies, finite fields, and computational tools, ordered from foundational definitions to advanced applications.
What This Cheat Sheet Covers
This topic spans 19 focused tables and 190 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
Table 1: Group FundamentalsTable 2: Subgroups, Cosets, and Lagrange's TheoremTable 3: Group Homomorphisms and IsomorphismsTable 4: Isomorphism TheoremsTable 5: Normal Subgroups and Quotient GroupsTable 6: Cyclic Groups and Group PresentationsTable 7: Symmetric and Alternating GroupsTable 8: Group Actions, Orbits, and StabilizersTable 9: Sylow TheoremsTable 10: Rings and IdealsTable 11: The Integral Domain HierarchyTable 12: Fields and Field ExtensionsTable 13: Finite Fields $\text{GF}(p)$ and $\text{GF}(p^n)$Table 14: Galois Theory and Solvability by RadicalsTable 15: Polynomial Rings and Irreducibility TestsTable 16: Modules over RingsTable 17: Key Theorems Quick ReferenceTable 18: Applications in Cryptography and Coding TheoryTable 19: Computational Algebra Tools and Gröbner Bases
Table 1: Group Fundamentals
| Concept | Example / Notation | Description |
|---|---|---|
(\mathbb{Z},+), (S_n,\circ) | Set G with binary operation satisfying closure, associativity, identity, and inverses | |
a,b\in G \Rightarrow ab\in G | Operation stays within the set | |
(ab)c=a(bc) | Grouping of operations does not matter | |
e\cdot a=a\cdot e=a | Unique element that leaves others unchanged | |
a\cdot a^{-1}=e | Each element has a two-sided inverse | |
(\mathbb{Z},+), (\mathbb{Z}/n\mathbb{Z},+) | Group where ab=ba for all elements | |
(S_n,\circ) for n\geq 3 | Group where commutativity fails for some elements |