Tour:Introduction two (beginners): Difference between revisions

Revision as of 13:38, 3 June 2008

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour)
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Hopefully, by the time you've reached this part of the guided tour, you have the basic definitions of a group and have some understanding of this definition. In this part, we'll see more about how to prove simple things about groups, and how to manipulate equations in groups.

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:

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

The goal of this part is to:

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

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 Hopefully, by the time you've reached this part of the guided tour, you have the basic definitions of a group and have some understanding of this definition. In this part, we'll see more about how to prove simple things about groups, and how to manipulate equations in groups.
-We'll begin by reviewing some simple examples that we've seen implicitly: equivalence of definitions, and some basic things about manipulating groups. After that, we'll go through some survey articles.
+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.
-The structure of groups is fairly rigid, because the axioms that control this structure are very strong: associativity, neutral element (identity element), and inverse element. To appreciate this, we'll explore a bit about variations on the structure,and how the powerful axioms of groups make them particularly well-behaved.
+We'll see the following pages:
+* [[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:Associative binary operation|Associative binary operation]]
+* [[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]]
+Prerequisites for this part: Material covered in part one, or equivalent. Basically, the definitions of group, subgroup, trivial group and Abelian group.
+The goal of this part is to:
+* 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