In this part, we dive deeper into the theory of groups.

| | | | created: 2014-02-20 22:37:53 | modified: 2019-08-04 08:19:38 | by: *bookofproofs*

## 1.**Definition**: Generating Set of a Group

## 2.**Definition**: Group Homomorphism

## 3.**Proposition**: Additive Subgroups of Integers

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

## 5.**Definition**: Cyclic Group, Order of an Element

## 6.**Example**: Examples of Cyclic Groups

## 7.**Proposition**: Finite Order of an Element Equals Order Of Generated Group

## 8.**Proposition**: Group Homomorphisms with Cyclic Groups

## 9.**Lemma**: Cyclic Groups are Abelian

## 10.**Lemma**: Subgroups of Cyclic Groups

## 11.**Proposition**: Subgroups of Finite Cyclic Groups

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

## 13.**Definition**: Conjugate Elements of a Group

## 14.**Definition**: Cosets

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

## 16.**Lemma**: Subgroups and Their Cosets are Equipotent

## 17.**Theorem**: Order of Subgroup Divides Order of Finite Group (Lagrange)

## 18.**Theorem**: Order of Cyclic Group (Fermat's Little Theorem)

## 19.Symmetry Groups

## 20.**Definition**: Normal Subgroups

## 21.**Lemma**: Factor Groups

## 22.**Lemma**: Group Homomorphisms and Normal Subgroups

## 23.**Theorem**: First Isomorphism Theorem for Groups

## 24.**Theorem**: Classification of Cyclic Groups

## 25.**Theorem**: Classification of Finite Groups with the Order of a Prime Number

## 26.**Definition**: Group Operation