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, 176 flashcards. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
A jump-to index of every table row in this cheat sheet.
An interactive map of every table and concept in this topic.
Table 1: Group Fundamentals
A group is the simplest rich structure in algebra: a set with one operation obeying four axioms — closure, associativity, identity, and inverses. These rows nail down those axioms and the vocabulary built straight on top of them, including the all-important order of a group and of its elements, and the abelian/non-abelian split that decides whether the operation commutes.
| 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 | ||
$\lvert S_3\rvert=6$ | • Cardinality $\lvert G\rvert$ • can be finite or infinite | ||
$\text{ord}(a)=n$ where $a^n=e$ | • Smallest positive $n$ with $a^n=e$ • divides $\lvert G\rvert$ | ||
${e}$ | Group containing only the identity | ||
$a+b$ vs $ab$; $0$ vs $e$ | • Convention • additive often for abelian groups | ||
$G\times H$ | New group with componentwise operation | ||
$N\rtimes H$ | Extension of $N$ by $H$ with $H$ acting on $N$ |