Tour:Confidence aggregator three (beginners)

Group
Consider the following questions:


 * 1) What is a group? What are the different ways of defining groups, and why 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?

At the end of part two, you had precise answers to (1) and (2), and hazy ideas for (3) and (4). Now, having seen more results on groups, and explored why these fail for monoids, quasigroups and other structures, you should have a better understanding of (3) and (4).

Questions that we haven't considered so far:


 * 1) How does one understand the structure of a specific group?
 * 2) Why do we really care about groups?

Subgroup
Consider the following list of questions:


 * 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?

We obtained clear answers to (1) and (2) in part two of the tour, and we've obtained partial answers to (3) and (4) in part three of the tour. We've concentrated on three things: intersections, cosets, and joins. We saw the interplay between these, in terms of the fact that cosets partition a group, have the same size, and yield to us the result called Lagrange's theorem. In the mind's eye test, we further saw how arithmetic data about the order of a group puts constraints on the subgroup structure, as well as on the nature of generating sets.

Abelian group
We haven't explored Abelian groups much. The main point of interest about Abelian groups is that all the notions of left and right that we've studied (such as left and right cosets) become equivalent for an Abelian group.

Another point we touched upon in the mind's eye test is that the property of a group being Abelian can be tested by checking whether all elements in a generating set for the group commute.

Trivial group
We haven't seen the trivial group much, apart from its definition. However, the trivial group has occurred as an important example of an extreme case subgroup to every group: in every group, the identity element gives the trivial subgroup, and this is contained in every subgroup. We'll see more about the trivial group later.

Part two overview
We review some key facts we saw in part two, and how they've been used in part three.

In part two of the guided tour, we introduced the notions of neutral element (identity element) in a general set with multiplication. We talked of associativity and inverses and the interplay between them. We talked about how the inverse map is involutive. Then, we talked about criteria to determine when a subset is a subgroup. Finally, we talked about manipulating equations in groups.

All these have been used in establishing basic facts about cosets. In the mind's eye test, we saw the importance of these by studying the breakdown of the theory of cosets when we look at semigroups and quasigroups.

Left and right neutral elements are equal, left and right inverses are equal
We've seen that:


 * 1) Any left and right neutral element (or multiplicative identity) must be equal
 * 2) If the multiplication is associative, then left and right inverses are equal

By now, you should feel confident of these statements, their proofs, and why they are important.

Associativity means forget parethesization
We've seen that for an associative binary operation, we can drop the parenthesization when writing products. By now, you should feel confident of writing and manipulating products as strings, without parentheses, but keeping in mind the order in which terms are written.

Inverse map is involutive
The statements:

$$(xy)^{-1} = y^{-1}x^{-1}, \qquad (x^{-1})^{-1} = x$$

were crucial to establishing the natural bijection between the left and right coset spaces of a subgroup. Philosophically, reversing the order of multiplication is equivalent to replacing every element by its inverse.

You should feel confident about this statement regarding the inverse map.

Multiplying elements and subsets
In the survey article on manipulating equations in groups, we talked of the notation for multiplying a subset and an element, and how to make sense of such equations. In part three, we saw the theory of left and right cosets, which used the notation. By now, you should feel somewhat comfortably with the idea of multiplying a subset and an element, and how such expressions can be manipulated.

Finite group, and order of a group
In part two, we introduced the notion of finite group, and order of a group. Some natural questions that could give rise to:


 * 1) How does the structure theory of finite groups differ from that of infinite groups?
 * 2) What information does the order of a group give about its structure, and the nature of its subgroups?

By now, you should have a partial idea of both (1) and (2).