# Tour:Confidence aggregator four (beginners)

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

## Part one overview

Consider the following questions:

1. What is a group? What are the different ways of defining groups, and how are they equivalent?
2. How can a group be identified, and how can one verify the group axioms?
3. How does one prove general results about groups? How do the three conditions for being a group play with each other?
4. What happens if we remove one or more of the axioms of groups? In what ways does the structural beauty get spoilt?
5. What does it mean for two groups to be the same?
6. Can groups be classified?

In parts two and three, you saw tentative answers to questions (1)-(4). Part four, which introduced the important notions of multiplication tables and cyclic groups, should have cemented your understanding of questions (1)-(4), and also given some glimpse into questions (5) and (6). For question (5), we have the notion of isomorphism of groups. For question (6), note that we have successfully completed a classification of groups with a generating set of size one: the cyclic groups are, upto isomorphism, either the group of integers or the group of integers modulo $n$ for $n$ equal tothe order of the group.

### Subgroup

1. What is a subgroup? What are the different ways of defining subgroups, and why are they equivalent?
2. How can a subgroup be identified? How can one test whether a given subset is a subgroup?
3. How does the nature of subgroups of a group control the nature of the group?
4. How does one use the fact that a given subset is a subgroup, to deduce further things?

In parts two and three, we obtained a partial picture answer to some of these questions. In part four, we explored the answers to these questions for a particular kind of group: the cyclic group. We showed that any subgroup of the group of integers is cyclic on its smallest positive element. Further, any subgroup of a cyclic group is cyclic. This shows a close relation between the nature of a group and the nature of its subgroups.

### Abelian group

The cyclic groups that we studied in part four are the simplest examples of Abelian groups. This is essentially because any two powers of the same element commute. In fact, the structure of finite Abelian groups, and more generally, of finitely generated Abelian groups, is closely related to the structure of cyclic groups. This is covered in detail in later parts.

### Trivial group

The trivial group occurs as an example of a cyclic group.

## Part three overview

### Subgroups and cosets

In part three, we introduced the notion of subgroup, left coset, and right coset. For an Abelian group, the left cosets are the same as the right cosets. For the group of integers $\mathbb{Z}$, the group of multiples of $n$ form a subgroup, and its cosets are the congruence classes.

What we did further was to equip the collection of congruence classes with a group structure using the group structure on $\mathbb{Z}$. This generalizes quite a bit, as we shall see in parts six and seven of the tour: the coset spaces corresponding to some particular kinds of subgroups (called normal subgroups) obtain the structure of a group, called the quotient group.

### Intersections and joins of subgroups

We saw how to take joins and intersections of subgroups of the group of integers: these correpsond to the operations of greatest common divisor and least common multiple.

### Lagrange's theorem

Lagrange's theorem says that the order of a subgroup divides the order of the group. We've seen that for a cyclic group of order $n$, there exists a unique subgroup of order $d$ for $d | n$. Thus, cyclic groups provide an example of Lagrange's theorem, as well as show its tightness: every divisor can occur as the order of a subgroup when the whole group is cyclic.

### Generating sets

We saw in the mind's eye test of part four that the subgroup generated by a subset is cyclic on the greatest common divisor of all the elements of that subset.