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 $⟨X⟩$, 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