## Abstract

**Definition 1.1**. A

*group G*is a non empty set in which it is defined a binary operation, i.e. a function:

$$ G\times G\to G $$

*ab*denotes the image of the pair (

*a, b*),

- i)
the operation is associative: (

*ab*)*c*=*a*(*bc*), for all triples of elements*a, b, c*∈*G*; - ii)
there exists an element

*e*∈*G*such that*ea = a = ae*, for all*a*∈*G*. This element is unique: if*e′*is also such that*e′a = a = ae′*, for all*a*∈*G*,*ea = a*implies, with*a = e′*, that*ee′ = e′*, and*a = ae′*implies, with*a = e*, that*ee′ = e*. Thus*e′ = e*; - iii)
for all

*a*∈*G*, there exists*b*∈*G*such that*ab = e = ba*.

## Keywords

Finite Group Cyclic Group Proper Subgroup Dihedral Group Klein Group
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Copyright information

© Springer-Verlag Italia 2012