Tour:Introduction one (beginners)

We're now about to get started with part one of the guided tour for beginners. Part one focuses on the definitions of group, subgroup, trivial group and Abelian group. The focus here is on definition understanding.

We'll see the following pages:


 * Group: Gives two equivalent definitions of group. Proof of equivalence of definitions, and closer study of definition, deferred for part two.
 * Abelian group: Defines abelian group, in terms of group.
 * Subgroup: Gives multiple equivalent definitions of subgroup. Proof of equivalence of definitions, and closer study of definitions, deferred for part two.
 * Trivial group: Defines trivial group.
 * Verifying the group axioms: Explores how to verify the group axioms and show that a given structure is a group.
 * Understanding the definition of a group: Discusses the importance of the various components/axioms of the definition of a group.

We'll also see some consolidation pages:


 * Factsheet one: Consolidates definitions, notations, and important observations of part one, also hinting at what's coming in part two.
 * Entertainment menu one: Gives a list of interesting survey articles and entertainment articles related to the basic ideas of groups, their role and importance.
 * Mind's eye test one: Has quick mental tests to cement understanding and recall of the definitions seen in part one.
 * Examples peek one: Gives a quick peek into some examples, through problems. Optional, can be skipped. Helps give a better hands-on feel of groups. Relies only on concepts introduced so far, plus knowledge from other branches of mathematics.

Prerequisites, goals and general suggestions
Prerequisites for this part:


 * An understanding of set-theoretic notation
 * A basic understanding of functions between sets, unary and binary operations, and relations

Desirables for this part: A knowledge and understanding of notions like commutativity, associativity, additive and multiplicative identity elements, in the context of number systems like the integers, rational numbers, real numbers. Goal of this part:
 * Provide a basic understanding of the definitions of group, subgroup, trivial group, and Abelian group
 * Provide the skill of determining whether a set with a binary operation, forms a group

In this part, we are not focusing on:

How to handle this part of the tour:
 * Explaining how to prove statements regarding groups (that'll be covered on parts two, three and more)
 * Giving a feel of what the various kinds of groups and subgroups are

If you are new to the notion of groups, then follow the pages of the tour in the indicated sequence. In each page, try to focus on understanding the definition section, and use content in the other sections only as a guide and not for complete understanding. Keep paper and pencil with you to note down any doubts you have on reading a particular page. It is likely that these doubts will get resolved as you proceed through the tour.

Remember that everything you see here will be revisited many times over the guided tour, from different angles, so do not panic if you are not getting a complete picture at this stage.

If you've already been through the tour once, or have studied the notion of groups before, then you can use the tour as a starting point, checking out the various links suggested in the tour, and going to the main pages. The pages shown in the tour are abridged versions of the wiki pages on the subject, and there's a link at the top to the main wiki page.

Book:Artin
The content here roughly corresponds to page 42 of Artin (Chapter 2, Section 1, Point (1.10)) and Page 44 of Artin (Chapter 2, Section 2, Point (2.1)). Unlike this guided tour, Artin's text does not directly plunge into the definition of a group, but rather begins by explaining (pages 38-42) the notion of law of composition as well as provides justification for associativity, inverses, and identity element.

Some material equivalent to content in Chapter 2, Section 1 of Artin can be found in the later articles of part one, including understanding the definition of a group. A more in-depth study of the axioms that underlie the definition of a group is reserved for part two of the guided tour. Thus, students are advised to attempt the exercises pertinent to these sections in Artin, only after completing part two of the guided tour.

Book:DummitFoote
The content here roughly corresponds to parts of Chapter 1, section 1.1 (Page 16-22) of the book by Dummit and Foote. However, a lot of the content in this portion of Dummit and Foote is closely related to what we'll be covering in part two of the guided tour, namely, an in-depth study of the axioms underlying group structure, and how to prove small things about groups. Thus, students are advised to attempt the exercises pertinent to these sections in Dummit and Foote, only after completing part two of the guided tour.

Rationale for plunging into the definition
It would have been perfectly possible to give some motivation for defining groups, before plunging into the definition. However, we've deliberately chosen to plunge directly into the definition, partly in order to prepare algebra students for the most standard way of learning: accept a definition, and then understand it. Having examples and motivations before plunging into each definition is not a luxury we can always afford.

That said, the importance of examples and motivation, and the importance of understanding the different components of and variations on a definition, cannot be underestimated. We revisit the definition of a group from many angles, including specific examples and general concerns, throughout the guided tour, in order to cement the definition.