Hypomnemata

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Abstract Algebra

3 min read

Fraleigh 7e · Chapters 1 – 15

Structure through axioms.

A single study path through introductory group theory, from binary operations and the group axioms through homomorphisms, factor groups, and simple groups. Each chapter page gives rigorous summaries, companion exercises, and counterexamples worth memorizing.

groups subgroups permutations cosets homomorphisms quotients

Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.

God made the integers; all else is the work of man. — Kronecker

Chapter Atlas

Fifteen chapters in five movements

The chapter order follows Fraleigh exactly. The five groups below trace a single arc: from raw sets to groups, through internal structure, into the counting and classification theorems, and finally to the quotient machinery that connects everything.

I. Foundations

Ch 1. Introduction and Examples Build intuition for algebraic structures through sets equipped with operations and axioms. Ch 2. Binary Operations Closure, associativity, identity, and inverses: the building blocks of algebraic structures. Ch 3. Isomorphic Binary Structures When two structures are the same in every way that matters. Ch 4. Groups The central definition: associativity, identity, inverses, and the first examples.

II. Subgroup Structure

Ch 5. Subgroups How smaller groups live inside larger ones, checked by the subgroup criterion. Ch 6. Cyclic Groups The simplest groups, generated by repeating a single element. Ch 7. Generators and Cayley Digraphs Visualize group structure through generators and make abstract algebra concrete.

III. Permutations

Ch 8. Groups of Permutations Every group is a permutation group: Cayley's theorem and concrete cycle notation. Ch 9. Orbits, Cycles, and Alternating Groups Classify permutations by parity and meet the alternating group.

IV. Counting and Structure

Ch 10. Cosets and Lagrange Partition a group into cosets and turn that partition into a divisibility law. Ch 11. Direct Products and Abelian Groups Build new groups from old ones and classify all finitely generated abelian groups. Ch 12. Plane Isometries All distance-preserving maps of the plane: translations, rotations, reflections, glide reflections.

V. Homomorphisms and Quotients

Ch 13. Homomorphisms Structure-preserving maps, their kernels and images: the bridge between groups. Ch 14. Factor Groups Quotient groups built from normal subgroups: when and why they work. Ch 15. Factor-Group Computations and Simple Groups Practical quotient computations and groups with no proper nontrivial normal subgroups.

Method

Fast Workflow

A single daily cycle designed for active recall and long-term retention.

Read and Rewrite

Read one chapter page and rewrite the main definitions by hand.

Solve Cold

Do two companion exercises with no notes open.

Drill on Mobile

Drill the same chapter in the flashcard deck on your phone.

Recall a Counterexample

End the day by recalling one counterexample from memory.

Because Fraleigh is copyrighted, these public pages do not reproduce long textbook exercise statements verbatim. Instead they give rigorous chapter summaries, original companion exercises, textbook drill prompts for your own copy, and famous counterexamples worth memorizing.

The detailed problem bank lives privately in your vault under private/Mathematics/Abstract Algebra.