# 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 Guided tour for beginners:Introduction one

This part provides some very basic, introductory definitions. We do not focus here on the example-oriented motivation for these definitions. The definitions provided are:

- group
- Abelian group
- Subgroup
- Trivial group
- Verifying the group axioms
- 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

Get started with Guided tour for beginners:Introduction two

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)

The goal of this part is to give comfort in simple manipulations involving groups.

Articles covered in this part are:

- Some variations of group
- Equality of left and right neutral element
- Equality of left and right inverses
- Equivalence of definitions of group
- Invertible implies cancellative
- Associative binary operation
- Finite group
- Subsemigroup of finite group is subgroup
- Sufficiency of subgroup criterion
- Manipulating equations in groups

Continue to the definition of a group

## Part three

Get started with Guided tour for beginners:Introduction three

Prerequisites: Part two or equivalent

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.

- Intersection of subgroups is subgroup
- Union of two subgroups is not a subgroup
- Left coset of a subgroup
- Left cosets are in bijection via left multiplication
- Right coset of a subgroup
- Left and right coset spaces are naturally isomorphic
- Lagrange's theorem
- Generating set of a group
- Subgroup generated by a subset
- Join of subgroups
- Some examples of groups and subgroups

## 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