# generating set of a group

Let $G$ be a group.

A subset $X\subseteq G$ is said to *generate* $G$
(or to be a *generating set ^{}* of $G$)
if no proper subgroup

^{}of $G$ contains $X$.

A subset $X\subseteq G$ generates $G$ if and only if
every element of $G$ can be expressed as
a product^{} of elements of $X$ and inverses^{} of elements of $X$
(taking the empty product to be the identity element^{}).
A subset $X\subseteq G$ is said to be *closed under ^{} inverses*
if ${x}^{-1}\in X$ whenever $x\in X$;
if a generating set $X$ of $G$ is closed under inverses,
then every element of $G$ is a product of elements of $X$.

A group that has a generating set with only one element
is called a cyclic group^{}.
A group that has a generating set with only finitely many elements
is called a finitely generated group.

If $X$ is an arbitrary subset of $G$,
then the subgroup^{} of $G$ *generated by* $X$, denoted by $\u27e8X\u27e9$,
is the smallest subgroup of $G$ that contains $X$.

The *generating rank* of $G$ is
the minimum cardinality of a generating set of $G$.
(This is sometimes just called the *rank* of $G$, but this can
cause confusion with other meanings of the term rank.)
If $G$ is uncountable, then its generating rank is simply $|G|$.

Title | generating set of a group |

Canonical name | GeneratingSetOfAGroup |

Date of creation | 2013-03-22 15:37:14 |

Last modified on | 2013-03-22 15:37:14 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 7 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20A05 |

Classification | msc 20F05 |

Synonym | generating set |

Related topic | Presentationgroup |

Related topic | Generator^{} |

Defines | generate |

Defines | generates |

Defines | generated by |

Defines | subgroup generated by |

Defines | generating rank |

Defines | closed under inverses |

Defines | group generated by |