Abstract Algebra

What this unit covers

The unit follows the usual path of an abstract algebra course. It starts with sets, binary operations and algebraic structures, moves through groups, subgroups and cyclic groups, and then reaches permutations, isomorphisms, group actions, rings and ideals. Nearly every item was made by GoConqr users, and the mix is mostly flashcards and mind maps.

Where to start

Begin with the flashcard deck Abstract Algebra Definitions. It runs from equivalence relations and binary operations through identity elements, isomorphisms, groups, abelian groups, cancellation laws and subgroups. Definitions come first in this subject, because every proof depends on them.

Next, open the mind map titled Abstract Algebra that links binary operations, semigroups, monoids, inverses, groups and homomorphisms. A second map of the same name adds sets, functions, permutations and group actions, and it shows why a structure that is not closed is not a binary operation.

Working through groups

1. Read the notes called Groups, which open with the definition of a group and examples.
2. Test yourself with the decks Ch. 4: Cyclic Groups and Ch. 6: Group Isomorphisms. They cover criteria for powers of an element, orders of elements, Cayley's Theorem and the steps for showing a function is an isomorphism.
3. Use Permutation Groups and Cayley Tables to practise with concrete examples.
4. Move on to Rings: Intro and Definitions, Rings: Homomorphisms and Rings: Ideals.

Using maps, the quiz and the flowchart

Mind maps give you the shape of a topic, and flashcards help you recall its details. Read a map first, then close it and try the matching deck. The quiz Abstract Algebra THEOREMS is a good check once you have been through the definitions. When a problem stumps you, the flowchart on how to approach a question gives you a way in.

Where it sits

This unit belongs to the Advanced Pure Mathematics strand of Mathematics. Teachers can pick out single decks or maps for a class, and students can mix and match by chapter.

Choose one deck today, work through it, and then try the quiz to see what has stuck.