# 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 ${\left\langle X\right\rangle}$, 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