# Tour:Pedagogical notes three (beginners)

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General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part

Goal of this part: We'll seek answers to the questions:

• What can we say about set-theoretic operations done on subgroup (like unions and intersections)?
• How does the nature of a group control the nature of possible subgroups?
• What is special about finite groups and subgroups thereof?

The third part of the tour is where the pace of material begins to pick up. Learners are expected to follow more elaborate lines of reasoning and take a more proactive role in checking the equivalence of multiple formulations as they go through the tour. In addition to introducing some new ideas, this part of the tour also serves to reinforce and apply the various ways of manipulating equations in groups introduced in part two.

Learners facing problems in this part of the tour are strongly advised to review the material in part two, and go over the mind's eye test of part two again. The ideas behind many of the techniques introduced in part three are present in the mind's eye test.

## Intersections of subgroups

### Main material

The tour begins with the statement and proof of the fact that an intersection of subgroups of a group is a subgroup. The statement is not hard to understand or prove, and its proof also helps review the criteria for being a subgroup. Learners unfamiliar with the notations used for indexing sets on infinite collections might feel a little intimidated.

### Reinforcement within the tour

The fact that an arbitrary intersection of subgroups is a subgroup surfaces later in the tour when the concept of subgroup generated by a subset is introduced: one of the definitions of the subgroup generated by a subset is as the intersection of all subgroups containing that subset.

### Reinforcement in the mind's eye test

The mind's eye test covers the idea in many different ways:

• In the section on intersections and joins, there are questions dealing with analogues of the statement for other algebraic structures, such as monoids and quasigroups. These questions help reinforce the proof ideas.
• There are questions combining the ideas of intersections of subgroups with the ideas of Lagrange's theorem and index. These seek to determined constraints on the order and index of the intersection of two subgroups in terms of the order and index of each of the subgroups.
• There is a question later asking for a proof that an intersection of left cosets, if nonempty, is a left coset of the intersection of the corresponding subgroups.

## Unions of subgroups

### Main material

The second page of this part is devoted to proving that the union of two subgroups is not a subgroup (unless one is contained in the other). This is in sharp contrast to the case of the intersection.

### Reinforcement within the tour

Later in the tour, the union of subgroups comes up again when the notion of join of subgroups is introduced: it is defined as the subgroup generated by their union.

### Reinforcement in the mind's eye test

The mind's eye test covers the idea in many different ways:

• In the section on intersections and joins, there are questions dealing with analogues of the statement for other algebraic structures, such as monoids and quasigroups. These questions help reinforce the proof ideas.
• There are also some additional questions on unions of subgroups: for instance, the fact that every group is a union of cyclic subgroups, and that a directed union of subgroups is a subgroup (only the case of a chain is asked as a question).

## Coset spaces

### Main material

A lot of important and relatively hard ideas are introduced here. Learners having difficulty with these ideas are strongly urged to review the tour's survey article on manipulating equations in groups and mind's eye test two, particularly the section involving left and right multiplication maps and the section involving subsets of groups.

The tour page on left coset introduces a number of equivalent definitions. Proving that these definitions are equivalent is a straightforward exercise in manipulating equations in groups, and learners are encouraged to do this as it makes the coming pages much easier to understand. Very similar questions are present in mind's eye test two.

After this, two important ideas are introduced: the left cosets are in bijection via left multiplication, and they partition the group. These ideas are hard to grasp, but a review of the general ideas for manipulating equations in groups can be very helpful.

After this, the dual notion of right coset is introduced, and a natural bijection between the left coset space and right coset space is introduced. With this, the index of a subgroup is defined, and Lagrange's theorem is introduced (more on Lagrange's theorem in the next section of these notes).

### Reinforcement in the mind's eye test

• In the subsection Subgroups and cosets, there are two basic questions reinforcing the definitions of left coset. These questions also help reinforce the multiple definitions of left coset and how those definitions are equivalent.
• Further down in the tour, the subsection Cosets and intersections reviews cosets from a somewhat different perspective.