# Discovering Group Theory

## A Transition to Advanced Mathematics

## Tony Barnard, Hugh Neill

- 219 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android

# Discovering Group Theory

## A Transition to Advanced Mathematics

## Tony Barnard, Hugh Neill

## About This Book

Discovering Group Theory: A Transition to Advanced Mathematics presents the usual material that is found in a first course on groups and then does a bit more. The book is intended for students who find the kind of reasoning in abstract mathematics courses unfamiliar and need extra support in this transition to advanced mathematics.

The book gives a number of examples of groups and subgroups, including permutation groups, dihedral groups, and groups of integer residue classes. The book goes on to study cosets and finishes with the first isomorphism theorem.

Very little is assumed as background knowledge on the part of the reader. Some facility in algebraic manipulation is required, and a working knowledge of some of the properties of integers, such as knowing how to factorize integers into prime factors.

The book aims to help students with the transition from concrete to abstract mathematical thinking.

## Frequently asked questions

## Information

## 1

*Proof*

### 1.1 The Need for Proof

**Proof**

*n*is the smaller whole number. Then (

*n*+ 1) is the larger number, and their sum is

*n*+ (

*n*+ 1) = 2

*n*+ 1. Since this is one more than a multiple of 2, it is odd. â

*a*and

*b*are even numbers, then

*a*+

*b*is even.

**Proof**

*a*is even, then it can be written in the form

*a*= 2

*m*where

*m*is a whole number. Similarly

*b*= 2

*n*where

*n*is a whole number. Then

*a*+

*b*= 2

*m*+ 2

*n*= 2(

*m*+

*n*). Since

*m*and

*n*are whole numbers, so is

*m*+

*n*; therefore

*a*+

*b*is an even number. â

*a*+

*b*when

*a*and

*b*are not both even. It simply makes no comment on any of the three cases: (1)

*a*is even and

*b*is odd; (2)

*a*is odd and

*b*is even; and (3)

*a*and

*b*are both odd.

*a*+

*b*

**is**even in case (3) but the statement of Example 1.1.2 says nothing about case (3).

*P*then

*Q*,â where

*P*and

*Q*are statements such as â

*a*and

*b*are evenâ and â

*a*+

*b*is even,â you cannot deduce anything at all about the truth or falsity of

*Q*if the statement

*P*is not true.