Tour:Confidence aggregator two (beginners)

Part one overview
In part one of the guided tour, we saw four things: group, Abelian group, trivial group, and subgroup.

Group
Consider the following list of 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?

By now, you should have precise answers to (1) and (2), and a general picture of the answers to (3) and (4). Quickly review the answers to (1) and (2) and try to formulate answers to (3) and (4) as best as you can.

Questions that we haven't yet explored are:


 * 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?

We've obtained reasonable answers to (1) and (2). Review these answers. Questions that we haven't explored yet:


 * 1) How does the nature of subgroups of a group control the nature of the group?
 * 2) How does one use the fact that a given subset is a subgroup, to deduce further things?

Abelian group
We haven't explored Abelian groups much apart from their definition. To gain more confidence in Abelian groups, go through the entire tour, this time replacing group with Abelian group. In what sense does the picture change, or simplify?

Trivial group
We haven't explored the trivial group much, apart from its definition. We'll see more about it later.

Interplay
Think quickly about the interplay between:


 * Associativity and inverses
 * Inverses and cancellation
 * Subgroup and finite group