Groupprops, The Group Properties Wiki (pre-alpha)

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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.

Before starting, you should read the general instructions. You may also find it useful to read the pedagogical notes that explain the structure of the tour in more detail.

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:

• Some variations of group: Defines weaker notions than groups, where one or more of the axioms or conditions for a group is relaxed.
• Equality of left and right neutral element: A short, mind's eye proof.
• Equality of left and right inverses: A short, mind's eye proof.
• Equivalence of definitions of group: Consolidates the definition of group by proving how two apparently different definitions seen in part one are equivalent.
• Invertible implies cancellative: A short, mind's eye proof.
• Equivalence of definitions of subgroup: Consolidates the definition of group by proving how two apparently different definitions seen in part one are equivalent.
• Associative binary operation: Defines and discusses important aspects of associative binary operations.
• Inverse map is involutive: A short, mind's eye proof.
• Order of a group: A simple definition.
• Finite group: A simple definition.
• Subsemigroup of finite group is subgroup: Applies ideas seen previously to prove a simple result on subsets of finite groups.
• Sufficiency of subgroup criterion: Applies ideas seen previously to prove a certain criterion for checking whether a subset is a subgroup.
• Manipulating equations in groups: Explores how equations and expressions in groups are manipulated. Consolidates material seen in parts one and two.

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:

• Intersection of subgroups is subgroup: A short, mind's eye proof
• Union of two subgroups is not a subgroup: A somewhat long, but essentially mind's eye proof
• Left coset of a subgroup: A simple definition.
• Left cosets partition a group: A short, mind's eye proof.
• Left cosets are in bijection via left multiplication: A short, mind's eye proof.
• Right coset of a subgroup: A simple definition.
• Left and right coset spaces are naturally isomorphic: A short, mind's eye proof.
• Index of a subgroup: A simple definition.
• Lagrange's theorem: An important result for finite groups, with a short, mind's eye proof.
• Generating set of a group: A simple definition.
• Subgroup generated by a subset: A simple definition.
• Join of subgroups: A simple definition of subgroup generated or join.

We'll also see some consolidation pages:

• Factsheet three
• Mind's eye test three
• Confidence aggregator three
• Interdisciplinary problems three
• Examples peek three

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

Get started with Tour:Introduction three (beginners)

This part of the tour is a preliminary look at some important classes of examples of groups, specifically, cyclic groups. We also pack here some general tools and approaches that will be useful later on.

• Multiplication table of a finite group: A tabular representation of the multiplication rule of a finite group.
• Isomorphism of groups: A straightforward definition of what an equality of two groups would mean.
• Isomorphic groups: A straightforward definition of what it means for two groups to be equal.
• Group of integers
• Group of integers modulo n
• Order of an element
• Cyclic group: A cursory definition of cyclic group.
• Equivalence of definitions of cyclic group
• Every nontrivial subgroup of the group of integers is cyclic on its smallest element
• Subgroup containment relation equals divisibility relation on generators
• No proper nontrivial subgroup implies cyclic of prime order
• Exploration of cyclic groups: A survey article looking at cyclic groups, how they are constructed, and a number of interesting facts about them.
• Cyclicity is subgroup-closed
• Multiplicative group modulo n
• Elements of multiplicative group equal generators of additive group
• Multiplicative group modulo a prime is cyclic

Prerequisites for this part: Content covered in parts one, two, and three (or equivalent content). In particular, the definitions of group, subgroup, trivial group, Abelian group, identity element, inverses, intersection of subgroups, join of subgroups, generating set of a group, left coset of a subgroup. Also, the major facts proved about these. Goal of this part: The goal here is a preliminary study an important class of groups: the cyclic groups. We study these from the viewpoint of how they occur naturally, and from the viewpoint of the generic tools we've developed for handling groups and subgroups.