Tour:Pedagogical notes one (beginners)

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This page is a Pedagogical notes page, part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part

This page has pedagogical notes giving an explanation of the content, scope and organization of part one of the guided tour. We focus on the four concepts introduced here, and the follow-up to each of these in later parts.

Goal of this part:

  • Provide a basic understanding of the definitions of group, subgroup, trivial group, and Abelian group
  • Provide the skill of determining whether a set with a binary operation, forms a group

In this part, we are not focusing on:

  • Explaining how to prove statements regarding groups (that'll be covered on parts two, three and more)
  • Giving a feel of what the various kinds of groups and subgroups are

Group

Definition

Two definitions of group are presented. One definition starts by assuming only a binary operation, and describes the identity element and inverses as additional conditions on this binary operation. In the other definition, the binary operation, identity element and inverses are considered part of the group structure.

The difference between these definitions may not be easy to spot for first-time learners (and it may be missed by experienced mathematicians as well, since the two definitions are obviously equivalent for them). Thus, putting the two definitions is not intended to produce an immediate reaction in learners, but rather, to introduce them to the fact that there are two slightly different definitions. The difference between the definitions, and why they are equivalent, is presented in part two.

Notation

The typical notational conventions for a group are introduced here. Again, it is not expected that first-time learners will completely absorb all these pieces of notation. The primary purpose of introducing this notation is to give learners a first-time introduction to the notation. The notation is reinforced in the many subsequent pages.

Examples

The examples given in the main part of the page (before the WHAT'S MORE) rely on basic concepts like integers, rational numbers and reals. These examples do not form a representative range: one common defect to them is that they are all Abelian. However, they illustrate aspects like existence of identity elements and existence of inverses (when contrasted with the accompanying non-examples).

The examples after the WHAT'S MORE cover a more representative range of examples. However, these examples may require some effort and/or prerequisites to comprehend, and are best read during a second round of the tour.

Checking that something is a group

This is discussed in some detail in the tour page: Tour:Verifying the group axioms, along with some illustrative examples. This should give learners a reasonable introductory idea of how to determine whether something is a group, although a better intuition in the matter can only come later.

Understanding the definition better

The tour page Tour:Understanding the definition of a group looks at the various components of the definition of a group, why they are there, and why being a group is a good thing. While this page does not give a complete explanation of why groups are important, it can help resolve the initial curiosity of learners.

Tolerance thresholds for learners

This particular style of definition may leave learners uncomfortable with regard to a number of things:

  • Why define groups in this way? Why define groups at all?
  • Why not assume commutativity for groups?
  • What are the examples of groups?
  • What is the importance of groups?
  • Why give multiple, seemingly equivalent, definitions of group? Why are they equivalent?

The last two tour pages, which are on verifying the group axioms and understanding the definition of a group, help resolve some of these. However, not all these concerns are adequately addressed in part one of the tour. Thus, it is best for learners not to expect a complete explanation at the end of part one.

To help address learners' curiosity, there is a separate entertainment menu page on the tour: Tour:Entertainment menu one (beginners). This links to other pages dealing with the history of groups, groups as symmetry, and some fun pages about groups.

Also, learners are promised that the concepts will be explored in more detail in part two of the tour, allowing them to defer the resolution of discomfort.

Subgroup

Definition

Three definitions of subgroup are presented. The first definition simply talks of a subset closed under the binary operation of the group, and requires the subset to form a group under the binary operation. The second definition talks of a subset closed under all the three group operations: the binary operation, the inverse map, and the identity element. The third definition is in terms of the subgroup criterion.

Learners are expected to go through all definitions, particularly understanding the first two. Again, it is unlikely that first-time learners will appreciate the difference between the definitions, but it is possible that seeing these two equivalent definitions of subgroup helps learners understand the difference between the two definitions of group. The third definition, involving the subgroup criterion, definitely looks different, and learners are not expected to immediately see why it is equivalent to the first two definitions. The equivalence of these definitions is presented in part two.

Notation

The notation for a subgroup is fairly straightforward, and learners are expected to grasp and remember it.

Examples

The examples before the WHAT'S MORE part are not intended as a representative range, but rather, simply to give a quick idea of what a subgroup means. As in the case of group, these examples do not cover non-Abelian group. The examples after WHAT'S MORE include non-Abelian groups.

Tolerance thresholds for learners

This particular style of definition may leave learners uncomfortable with regard to a number of things:

  • Why define subgroups in this way? What is the purpose of defining subgroups at all?
  • Why give multiple, seemingly equivalent, definitions of subgroup? Why are they equivalent?

These concerns are not addressed in part one of the tour, except for a brief mention of this in the tour page on verifying the group axioms, where it is pointed out that it may be easier to prove that a subset of a group forms a subgroup under the induced operations, than prove from scratch that it is a group. That's because we only need to show closure and do not need to show associativity.

Also, learners are promised that the concepts will be explored in more detail in part two of the tour, allowing them to defer the resolution of discomfort.

Abelian group

Definition

A definition of Abelian group is presented, as a group where any two elements commute. The importance of Abelian groups is cleatr to learners as all examples of groups provided before the WHAT'S MORE section are Abelian.

Tolerance thresholds for learners

Learners may have the following natural questions:

  • How does the theory of groups and Abelian groups differ? Why create a separate notion for Abelian group?

This is not resolved in this part of the tour.