## Group Theory

Above, we have learned about magmas, semigroups, monoids as simple types of algebraic structures. In this chapter, we will introduce a more complex algebra – the *group*. We continue with our tabular overview to indicate, which properties of a group fulfills:

Algebra $(X,\ast)$ | Closure | Associativity | Neutral Element | Existence of Inverse | Cancellation | Commutativity |
---|---|---|---|---|---|---|

Magma | required | (✔) | (✔) | (✔) | (✔) | (✔) |

Semigroup | required | required | (✔) | (✔) | (✔) | (✔) |

Monoid | required | required | required | (✔) | (✔) | (✔) |

Group | required | required | required | required | ✔ | (✔) |

We will see later that, in every group, the existence of inverse elements already ensures the cancellation property. Therfore the entry is “✔” and not the optional “(✔)”.

Groups are so important structures in algebra that *group theory* can be concidered as a separate branch of algebra and mathematics. The results of group theory have various applications in physics and technology.

| | | | created: 2014-02-20 22:37:53 | modified: 2019-02-10 15:26:40 | by: *bookofproofs*

## 1.**Definition**: Group

## 2.**Definition**: Group Order

## 3.**Definition**: Commutative (Abelian) Group

## 4.**Motivation**: Calculations in a Group

## 5.**Example**: Examples of Groups

## 6.**Definition**: Subgroup

## 7.**Theorem**: Construction of Groups from Commutative and Cancellative Semigroups

## 8.Symmetry Groups

## 9.**Definition**: Group Homomorphism

## 10.**Proposition**: Properties of Cosets

## 11.**Definition**: Direct Product of Groups

## 12.**Definition**: Cosets

## 13.**Definition**: Cyclic Group

## 14.**Definition**: Group Operation

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