Difference between revisions of "Tour:Getting started (beginners)"

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{{quotation|Get started with [[Tour:Introduction one (beginners)]]}}
 
{{quotation|Get started with [[Tour:Introduction one (beginners)]]}}
This part provides some very basic, introductory definitions. We do not focus here on the example-oriented motivation for these definitions. The pages are:
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{{#lst:Tour:Introduction one (beginners)|pagelist}}
 
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{{#lst:Tour:Introduction one (beginners)|prerequisite}}
* [[Guided tour for beginners:Group|group]]
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{{#lst:Tour:Introduction one (beginners)|goal}}
* [[Guided tour for beginners:Abelian group|Abelian group]]
 
* [[Guided tour for beginners:Subgroup|Subgroup]]
 
* [[Guided tour for beginners:Trivial group|Trivial group]]
 
* [[Guided tour for beginners:Verifying the group axioms|Verifying the group axioms]]
 
* [[Guided tour for beginners:Understanding the definition of a group|Understanding the definition of a group]]
 
Prerequisites for this part:
 
 
 
* An understanding of set-theoretic notation
 
* A basic understanding of functions between sets, unary and binary operations, and relations
 
 
 
The goal of this part is to:
 
 
 
* 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
 
 
 
 
==Part two==
 
==Part two==
  
{{quotation|Get started with [[Guided tour for beginners:Introduction two]]}}
+
{{quotation|Get started with [[Tour:Introduction two (beginners)]]}}
  
 
This part focuses on providing an understanding of how to do simple manipulations involving groups. We begin by generalizing some of the ideas involving groups, and discussing proofs involving some of the basic manipulations.
 
This part focuses on providing an understanding of how to do simple manipulations involving groups. We begin by generalizing some of the ideas involving groups, and discussing proofs involving some of the basic manipulations.
  
Prerequisites: Part one (or equivalent)
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{{#lst:Tour:Introduction two (beginners)|pagelist}}
 
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{{#lst:Tour:Introduction two (beginners)|prerequisite}}
The goal of this part is to give comfort in simple manipulations involving groups.
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{{#lst:Tour:Introduction two (beginners)|goal}}
 
 
Articles covered in this part are:
 
 
 
* [[Guided tour for beginners:Some variations of group|Some variations of group]]
 
* [[Guided tour for beginners:Equality of left and right neutral element|Equality of left and right neutral element]]
 
* [[Guided tour for beginners:Equality of left and right inverses|Equality of left and right inverses]]
 
* [[Guided tour for beginners:Equivalence of definitions of group|Equivalence of definitions of group]]
 
* [[Guided tour for beginners:Invertible implies cancellative|Invertible implies cancellative]]
 
* [[Guided tour for beginners:Equivalence of definitions of subgroup|Equivalence of definitions of subgroup]]
 
* [[Guided tour for beginners:Associative binary operation|Associative binary operation]]
 
* [[Guided tour for beginners:Inverse map is involutive|Inverse map is involutive]]
 
* [[Guided tour for beginners:Associative binary operation|Associative binary operation]]
 
* [[Guided tour for beginners:Order of a group|Order of a group]]
 
* [[Guided tour for beginners:Finite group|Finite group]]
 
* [[Guided tour for beginners:Subsemigroup of finite group is subgroup|Subsemigroup of finite group is subgroup]]
 
* [[Guided tour for beginners:Sufficiency of subgroup criterion|Sufficiency of subgroup criterion]]
 
* [[Guided tour for beginners:Manipulating equations in groups|Manipulating equations in groups]]
 
 
 
 
==Part three==
 
==Part three==
  
{{quotation|Get started with [[Guided tour for beginners:Introduction three]]}}
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{{quotation|Get started with [[Tour:Introduction three (beginners)]]}}
 
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{{#lst:Tour:Introduction three (beginners)|pagelist}}
Prerequisites: Part two or equivalent
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{{#lst:Tour:Introduction three (beginners)|prerequisite}}
 
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{{#lst:Tour:Introduction three (beginners)|goal}}
This part focuses a bit more on subgroups; the notion of intersection and union of subgroups and whether a union of subgroups is a subgroup.
 
 
 
* [[Guided tour for beginners:Intersection of subgroups is subgroup|Intersection of subgroups is subgroup]]
 
* [[Guided tour for beginners:Union of two subgroups is not a subgroup|Union of two subgroups is not a subgroup]]
 
* [[Guided tour for beginners:Left coset of a subgroup|Left coset of a subgroup]]
 
* [[Guided tour for beginners:Left cosets partition a group|Left cosets partition a group]]
 
* [[Guided tour for beginners:Left cosets are in bijection via left multiplication|Left cosets are in bijection via left multiplication]]
 
* [[Guided tour for beginners:Right coset of a subgroup|Right coset of a subgroup]]
 
* [[Guided tour for beginners:Left and right coset spaces are naturally isomorphic|Left and right coset spaces are naturally isomorphic]]
 
* [[Guided tour for beginners:Index of a subgroup|Index of a subgroup]]
 
* [[Guided tour for beginners:Lagrange's theorem|Lagrange's theorem]]
 
* [[Guided tour for beginners:Generating set of a group|Generating set of a group]]
 
* [[Guided tour for beginners:Subgroup generated by a subset|Subgroup generated by a subset]]
 
* [[Guided tour for beginners:Join of subgroups|Join of subgroups]]
 
* [[Guided tour for beginners:Some examples of groups and subgroups|Some examples of groups and subgroups]]
 
  
 
==Part four==
 
==Part four==

Revision as of 17:20, 28 June 2008

Get started

We are about to get started on the guided tour for beginners. To get the most from this guided tour, stay faithful to it, i.e. read the articles in the order suggested. You will have various opportunities for detours: some other articles to read so as to get a better understanding of what you're touring, and some just for entertainment. Please try to open these detours in different windows/tabs so that you do not lose track of where you are in the main tour.

This tour is not intended to be a complete introduction to group theory, or a replacement for textbook or course materials. Rather, it is intended as a supplement. To get the most from this tour, keep open your main course book or lecture notes and make sure you can map what's there on the wiki, with what you're learning in the course or from the textbook.

The tour is structured as follows.

Part one

Get started with Tour:Introduction one (beginners)

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 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:

  • 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

Part two

Get started with Tour:Introduction two (beginners)

This part focuses on providing an understanding of how to do simple manipulations involving groups. We begin by generalizing some of the ideas involving groups, and discussing proofs involving some of the basic manipulations.


We'll see the following pages:

We'll also see some consolidation pages:

  • Factsheet two: Summarizes facts seen in parts one and two.
  • Confidence aggregator two: Asks questions to help the reader self-assess and introspect on what has been learned in parts one and two.
  • Mind's eye test two: Problems based on parts one and two, that help sharpen the mind's eye and consolidate material learned in these parts.
  • Examples peek two: Gives a quick peek into some examples, through problems. Develops further on the themes seen in examples peek one.
  • Interdisciplinary problems two: Problems related to other parts of mathematics. Optional, and recommended for people who have some famiiliarity with those other branches.


Prerequisites for this part: Material covered in part one, or equivalent. Basically, the definitions of group, subgroup, trivial group and Abelian group.

Desirables for this part: Experience with a few groups, like the additive group of real or rational numbers, and with some monoids, such as the additive monoid of nonnegative integers (knowing the abstract concepts of group and monoid isn't necessary).


Goal of this part:

  • Provide some intuition into how to manipulate the various conditions for being a group, to prove simple statements about groups
  • Give an idea of the way the axioms control and make rigid the structure of a group

Part three

Get started with Tour:Introduction three (beginners)

In this part, we'll see:

We'll also see some consolidation pages:

Prerequisites for this part: Parts one and two (or equivalent content)

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?


Part four

Not yet prepared

Part five

Not yet prepared