# group theory definition

## properties

Before introduce group definition, let me list those properties mentioned.

`Closure`

:

For all $a, b \in G$, the relation $a \bullet b \in G$ holds.`associative:`

For all $a, b, c \in G$, the equation $(a \bullet b) \bullet c = a \bullet (b \bullet c)$ holds.`Commutative`

For all $a, b \in G$, $a \bullet b = b \bullet a$.`Identity element`

:

There exists an element $e \in G$, such that for all elements $a \in G$, the equation $e \bullet a = a$ holds.`Inversible`

:

For each $a \in G$, there exists an element $a^{-1} \in G$ such that $a \bullet a^{-1} = e$, where $e$ is the identity element.`Distributive`

:

For all $a, b, c \in G$, the equation $(a+b) \bullet c=a \bullet c+b \bullet c$ holds.

## group(G,•)

consists of a set of elements together with an operation $\bullet$ such that:

property | $\bullet$ | |
---|---|---|

clojure | $\checkmark$ | |

associative | $\checkmark$ | |

identity | $\checkmark$ | |

inversible | $\checkmark$ | |

commutative | $\times$ | |

distributive | $\times$ |

## abelian group (A, •)

Abelian group is a `group`

, but also have one additional property:

property | $\bullet$ | |
---|---|---|

clojure | $\checkmark$ | |

associative | $\checkmark$ | |

identity | $\checkmark$ | |

inversible | $\checkmark$ | |

commutative | $\checkmark$ | |

distributive | $\times$ |

## rings

A commutative ring with unity $(R,+,*)$ is an algebraic structure consisting of a set of elements R together with two binary operations denoted `+`

and `*`

which satisfy the follow properties for all elements in `R`

:

property | $+$ | $*$ |
---|---|---|

clojure | $\checkmark$ | $\checkmark$ |

associative | $\checkmark$ | $\checkmark$ |

commutative | $\checkmark$ | $\checkmark$ |

identity | $\checkmark$ | $\checkmark$ |

inversible | $\checkmark$ | N/A |

distributive | $\times$ | $\checkmark$ |

## ideal ring

Let $(R,+,*)$ be a ring; A non-empty subset $I$ of $R$ called a `ideal`

of the ring if:

- $(I,+)$ is a group.
- $i*r \in I$ for all $i \in I$ and $r \in R$.

property | $+$ | |
---|---|---|

clojure | $\checkmark$ | |

associative | $\checkmark$ | |

identity | $\checkmark$ | |

inversible | $\checkmark$ | |

commutative | $\times$ | |

distributive | $\times$ |

## field

field is a set of elements which is closed under two binary operations, which we denote by $+$ and $\times$.

property | $+$ | $\times$ |
---|---|---|

clojure | $\checkmark$ | $\checkmark$ |

associative | $\checkmark$ | $\checkmark$ |

commutative | $\checkmark$ | $\checkmark$ |

identity | 0 | 1 |

inversible | $\checkmark$ | !0 |

distributive | $\times$ | $\checkmark$ |