# Tour:Introduction two (beginners)

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Inquiry problems one (beginners)|NEXT: Some variations of groupPREVIOUS SECTION: Introduction one|NEXT SECTION: Introduction three

General instructions for the tour | Pedagogical notes for the tour

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:

- 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, goal and general suggestions

**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

**Some general suggestions/motivation for this part**: So far, you have seen the definition of group, subgroup, trivial group and Abelian group. In this part, we focus on some simple facts about groups and how to prove them.

The purpose of this part is two-fold. First, we establish a lot of basic facts about groups that will help you understand the importance of the concept of groups and how to think in terms of groups. Second, the methods used to prove these facts give you an idea of how to go about proving things related to groups.

At first glance, neither the statements nor the proofs may seem clearly motivated. However, once you have a reasonable understanding of groups, all the statements in this part should be very clear and very intuitive. Also, the approach used here is deliberately a *minimalistic* approach -- for each fact, the minimal hypotheses needed to establish that fact are chosen, so that you can see precisely what conditions lead to what theorems. It is also strongly recommended that you do 'all* the Mind's Eye Test problems for this part, to consolidate the material covered here.*

You can learn more about the pedagogical approach of this part of the tour at Tour:Pedagogical notes two (beginners).

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Inquiry problems one (beginners)|NEXT: Some variations of groupPREVIOUS SECTION: Introduction one|NEXT SECTION: Introduction three

General instructions for the tour | Pedagogical notes for the tour