Book which fleshes theory by case studies in abstract algebra

Question

I want to enrich my intuition of the structures studied in Abstract Algebra by going through case studies of them. So, for example, I want some example to be worked through, and in process, the theory to be developed for it.

For example, something like page-298 of Gallian Helpful (section on weird-dice). Also, I liked the sections on graph theory by Kennth H Rosen in his book of discrte mathematics.

Answer 1

If I am correctly understanding your request, you are looking for ways to make abstract algebra theory computational to build intuition and experience.

There are many resources that have been created to support this using Computer Algebra Systems.

Examples include professional packages like Magma, Mathematica, Maple and open source packages like GAP, Pari/GP, FriCAS, SAGE and others.

Many books and course materials have been written to go along with these systems and to help provide a more intuitive, exploratory, and experimental approach to abstract algebra and any math area. Examples include

• Abstract Algebra, An Interactive Approach - Second Edition (many wonderful examples to further explore using SAGE and others) by William Paulsen

• http://abstract.pugetsound.edu/sage-aata.html (each section has SAGE exercises and examples), also see https://doc.sagemath.org/html/en/prep/Quickstarts/Abstract-Algebra.html

• Algebra for Applications (uses GAP) by Arkadii Slinko

• Group Explorer - https://nathancarter.github.io/group-explorer/index.html

• Learning Abstract Algebra with GAP - https://mathstat.slu.edu/~rainbolt/FullManual8th.pdf

• http://www2.macaulay2.com/Macaulay2/ - used to study polynomial rings, especially Gröbner bases

• SINGULAR - used to study polynomial rings

• CoCoA System - for Computations in Commutative Algebra

• Applications of Abstract Algebra with Maple and MATLAB, Second Edition by Richard Klima, Neil Sigmon, Ernest Stitzinger

• Algebra, A Computational Introduction by John Scherk uses Mathematica and Maple

• Abstract Algebra: A Computational Approach by Charles C. Sims recommends using APL

• Visual Group Theory by Nathan Carter

• Geometries and groups, by Nikulin and Shafarevich

• Groups and Symmetry by Mark A. Armstrong

• Abstract Algebra: Examples and Applications by Justin Hill, Chris Thron, et al.

• Abstract Algebra: An Inquiry Based Approach by Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom

• ...many others

There are also useful resources to have handy like

• For group theory, Tim Dokchitser has a database called GroupNames. This provides presentations, important subgroups and quotients, and character tables.

• Group Tables and Subgroup Diagrams

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• Differences between GAP and PARI/GP?

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• ...and others.