Tour:Pedagogical notes two (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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Goal of this part:

  • Provide some intuition into how to manipulate the various conditions for being a group, to prove simple statements about groups
  • Give an idea of the way the axioms control and make rigid the structure of a group


Group

The main purpose of this part of the guided tour, with respect to the definition of group, is to give learners a clear understanding of how the different parts of the definition of group fit together and why each component is important.

For this purpose, some variations of the notion of group are introduced, including: maga (a set with a binary operation), semigroup (A set with an associative binary operation), monoid (a set with an associative binary operation having a two-sided identity element). Similarly, notions of neutral element/identity element, invertible and cancellative element are introduced.

Identity elements and the way they behave

The first fact page of the tour shows learners the statement that if a magma (set with binary operation) has a left neutral element and a right neutral element, they are both equal. While this is an extremely simple statement, the purpose of this page is to set the tone for learners, and make learners pause and think oabout the interplay between the binary operation and neutral elements. Learners are also encouraged to see for themselves how this implies that a binary operation can have at most one two-sided neutral element.

Some pedagogical comments:

  • There is an extraordinary amount of parsing and reduction here. Rather than simply prove that the identity element of a group is unique, this gives a statement in a much greater generality. The purpose of this is to illustrate to learners that some statements can be viewed in substantially greater generality than we ordinarily perceive them. Of course, it is unlikely that learners will recognize this consciously.
  • An entire page is devoted to a fact that would otherwise seem trivial. The idea is to impress upon learners that some facts are important, not because of their degree of difficulty, but because they control the language and framework of the subject. Again, it is unlikely that learners will recognize this consciously.

Inverses and the way they behave

Following closely on the heels of equality of left and right neutral elements is the statement of equality of left and right inverses with respect to a neutral element. Learners are encouraged to make careful note of the use of associativity, and are provided a justification for the manipulatino behind the proof. Learners are also encouraged to see how this implies that an element can have at most one two-sided inverse.

Some pedagogical comments:

  • The similar pattern between this fact and the previous one, as well as the similar way in which both facts give rise to correspond facts about "two-sided" constructions, emphasizes the similarity and common pattern across mathematical results. It is unlikely that learners will recognize this consciosuly.
  • The key difference between the facts, namely, the use of associativity in one case, also serves to highlight the importance of the axiomatic assumptions made in proving basic things. This might deepen learners' appreciation of associativity.

Definition equivalences for group and subgroup

The equivalence of definitions of group simply pieces together the newly learned facts with the definitions learned in part one. This serves as a double-revision as well as an exercise in piecing together ideas.

The equivalence of definitions of subgroup follows the establishment of the property that invertible elements can be cancelled -- another application of associativity. This equivalence of definitions further cements learners' understanding of the basic manipulations in groups.

Some pedagogical comments:

  • It is likely that the similarity as well as difference between multiple definitions of group, as well as between multiple definitions of subgroup, will start to click by this stage in the learner's minds.
  • Learners might also have a greater appreciation of the interplay between associativity, identity element and inverses by this stage.
  • By this time, learners would also have caught on to the theme of this part of the tour.

More on manipulating expressions

After the equivalence of definitions for group and subgroup, the focus shifts to establishing other useful properties necessary to manipulate expressions in groups.

Associative binary operation and reversal law

The page on associative binary operation summarizes more facts about associativity, repeating the fact that associativity allows us to drop parentheses when writing out an expression. The next page, about the involutive nature of the inverse map and about the reversal law, again points out the intricate link between associativity and inverses and gives ideas about manipulating expressions.

Some pedagogical comments:

  • By this stage in the tour, learners should get a sense of what can and cannot be proved using associativity, and how.

Finite groups, subsets closed under multiplication

The focus now shifts to techniques of manipulation for finite groups. This is begun by introducing definitions for finite group and order of a group, and then proceeding to prove that any nonempty multiplicatively closed subset of a finite group is a subgroup.

The proof of this result is a first of its kind in the tour. It involves a complex, more exploratory argument than the straightforward arguments shown so far. Specifically, it is the first time the idea of pick an element and take its powers is used.

Some pedagogical comments:

  • This page is expected to jolt learners into thinking about how, given a certain situation in a group, one can start reasoning with it: pick elements, multiply them.
  • Learners are also exposed to the idea of the pigeonhole principle: in a finite set there are bound to be repetitions and these repetitions can be combined with the structure of groups to prove important things.
  • The proof also gives a checklist-style revision of what it means to be a subgroup, and hence reinforces the concept of subgroup in the learners' minds.

This is followed by a page of the sufficiency of subgroup criterion. This page has a somewhat more complicated statement, but the proof follows a very similar template. Learners should not experience problems grasping this, and it should reinforce some of the lessons of the previous page. It also highlights how inverses can be manipulated.

Manipulating equations in groups

This page summarizes techniques for manipulating equations in groups, and thus equips learners with the tools needed to start with equations and modify them in various ways.

Some pedagogical comments:

  • Since learners have seen so many simple but painstaking examples of manipulation, as well as the rationale for these manipulations, they have the correct background to appreciate and register the content of the page.
This page is a Pedagogical notes page, part of the Groupprops guided tour for beginners (Jump to beginning of tour)
UP: Introduction two |
PREVIOUS SECTION Pedagogical notes: Pedagogical notes one|NEXT SECTION Pedagogical notes: Pedagogical notes three
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part