Tour:Pedagogical notes (beginners)
This page gives an overall description of the guided tour for beginners, along with pedagogical notes for instructors. Detailed notes on the individual parts of the tour, as well as the mind's eye tests, confidence aggregators and other end-of-part pages is found elsewhere.
Contents
The scope of the guided tour
The scope
This guided tour is intended to provide a reasonable complete coverage of introductory topics in group theory, as may be covered, for instance, in a:
- Single quarter of beginning undergraduate course in group theory
- Part of a single semester of beginning undergraduate course in algebra
- Beginning part of a single quarter/semester of an advanced undergraduate course in group theory or algebra
The specific knowledge and skills promised
Definitions
At the end of the guided tour, a learner should be able to give definitions of all the terms in Category:Basic definitions in group theory.
For the following terms, the learner should be able to:
- Give a precise definition
- Prove equivalence of all the typical definitions
- Provide a representative range of examples
- Judge whether an object handed to them satisfies the definition
- Apply the various components of the definition to straightforward problems
- Explain why every part of the definition is important
These terms are: group, subgroup, trivial group, Abelian group, cyclic group, isomorphism of groups, isomorphic groups, order of a group
For the following terms, the learner should be able to:
- Give a precise definition
- Prove equivalence of all the typical definitions
- Provide a few ilustrative examples
- Judge, in simple cases, whether an object handed to them satisfies the definition
- Apply the various components of the definition to a few straightforward problems
These terms are: left coset of a subgroup, right coset of a subgroup, normal subgroup, quotient map, index of a subgroup, homomorphism of groups, characteristic subgroup, group action, center, commutator subgroup, congruence on a group, conjugacy class, automorphism of a group, endomorphism of a group, external direct product, inner automorphism, internal direct product, maximal subgroup.
Facts
At the end of the guided tour, the learner should be able to recall quickly the proofs of facts that:
- Test multiple equivalent definitions of basic terms
- Apply directly one or more of the basic components of the definition
- Are equivalent to, or reformulations of, any of the important theorems, which include left cosets partition a group, left cosets are in bijection via left multiplication, Lagrange's theorem, the four isomorphism theorems (first, second, third, fourth).
Examples
The learner should feel comfortable taking any given group to start with, and applying the various techniques studied so far to ask and answer questions about it. This includes, for instance:
- For a finite group, using the order of the group to get information about the possible orders of subgroups, using Lagrange's theorem, as well as the pattern of intersection and union of these subgroups.
- PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
The order and organization
The guided tour is organized into ten parts. Each part has a particular focus. The organization is such that important primitive concepts are introduced earlier and revised more often, while the less important or more advanced ideas are introduced later and repeated less often. The gap between repetitions has been chosen carefully to maximize assimilation.
Here is a quick description of the parts:
- Part one (more at Tour:Pedagogical notes one (beginners)): Introduces the basic definitions of group, subgroup, trivial group and Abelian group, with a few examples, and an explanation of how to verify whether something is a group. The aim of this part is to give people basic familiarity with the definition. This part does not attempt to give any proofs.
- Part two (more at Tour:Pedagogical notes two (beginners)): Introduces some variations of the concept of group, including monoid, semigroup, and magma. These variations are not in themselves central; their main role is to emphasize what components of the definition of group are used in various proofs involving group. Walks through the proofs of uniqueness of identity element and inverses, as well as other basic results about necessary and sufficient conditions for being a subgroup. Reiterates the definitions of group and subgroup by proving equivalence of definitions. This part thus cements an understanding of the basic definitions. It does not attempt to give examples or further motivation.
- Part three (more at Tour:Pedagogical notes three (beginners)): Looks at the set-theoretical operations on subgroups in a group (intersection and union) and proves results about these. Looks at the concept of left and right cosets, and proves results leading to Lagrange's theorem. Introduces the concepts of generating set and subgroup generated. This part thus helps learners understand the set-theoretic structure of groups and subgroups and provides further practice in proving results about groups and subgroup, using the definitions of group and subgroup.
- Part four (more at Tour:Pedagogical notes four (beginners)): Develops an important class of examples of groups: cyclic groups. In the process, introduces the important notion of isomorphism of groups, multiplication tables, order of element, the role of cyclic subgroups, and other tools for the general study of finite groups. This part thus reviews all the basic facts established about groups while also giving concrete examples (Abelian ones) for learners to work with.
- Part five (more at Tour:Pedagogical notes five (beginners): Develops an important class of examples of groups: symmetric groups. In the process, introduces the important notion of external direct product of groups, group action on a set, cycle decomposition for permutations, notion of conjugacy and how it is worked out for permutations. The part reviews all the basic facts established about groups, and for the first time systematically explores non-Abelian examples.
- Part six (more at Tour:Pedagogical notes six (beginners)): Introduces homomorphism of groups, normal subgroups, kernels of homomorphisms, and the isomorphism theorems. Uses examples from cyclic and symmetric groups to cement this understanding. Carefully analyzes the quotient map. This part thus introduces some of the central ideas of group theory (homomorphisms, quotients) and related them both to past results and to past examples.
- Part seven (more at Tour:Pedagogical notes seven (beginners)): Introduces notions of endomorphism and automorphism of a group, views normality in terms of inner automorphism, studies metaproperties of normality (closed under joins, intersections, satisfies intermediate subgroup condition). Introduces related idea of characteristic subgroup, establishes relation between characteristicity and normality. Introduces center and commutator subgroup and shows that they are both characteristic. This part gets into the groove of subgroup properties, while continuing to give a better feel for the notion of normality developed in part five.
- Part eight (more at Tour:Pedagogical notes seven (beginners)): Introduces the general idea of a group action, views a group acting on itself in different ways, develops group action on coset space, fundamental theorem of group actions, class equation of a group, conjugacy classes, centralizers, normalizers, Cayley's theorem. Reviews past results like Lagrange's theorem. This part simultaneously introduces a very important paradigm in group theory and revises all the previous content in a new light.
- Part nine (more at Tour:Pedagogical notes eight (beginners)): Introduces internal direct products, looks at examples of these in cyclic and symmetric groups, discusses the structure theory of finitely generated Abelian groups. Introduces retracts, product of subgroups, and permutable complements. Studies examples like dihedral groups and generalized quaternion group.
- Part ten (more at Tour:Pedagogical notes nine (beginners)): Introduces concepts of normal core and normal closure. Proves results about index of a subgroup, both in finite and infinite groups. Proves results about finite subgroups in infinite groups. Combines ideas of group actions and other ideas previous studied.
- Part eleven (more at Tour:Pedagogical notes ten (beginners)): Introduces counterexamples to some plausible statements (for instance, gives a permutable subgroup that is not normal, and a non-Abelian group all whose subgroups are normal). Delves into methods to study specific finite groups. Introduces ideas of simple group and directly indecomposable group and proves the simplicity of .