# Tour:Getting started (beginners)

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:

- 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.
- Nonempty finite subsemigroup of 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

*Not yet prepared*

- Homomorphism of groups
- Isomorphism of groups
- Isomorphic groups
- Endomorphism of a group
- Automorphism of a group
- Automorphism group
- Inner automorphism
- Kernel
- Normal subgroup
- Quotient group
- First isomorphism theorem
- Second isomorphism theorem
- Third isomorphism theorem

## Part five

*Not yet prepared*

- Characteristic subgroup
- Characteristic implies normal
- External direct product
- Internal direct product
- Group property
- Subgroup property
- Subgroup-defining function
- Center
- Characteristic of normal implies normal
- Commutator subgroup
- Normality is strongly intersection-closed
- Normality is strongly join-closed
- Invariance property