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# 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 $|G|$.

## Mathematics Subject Classification

20A05*no label found*20F05

*no label found*

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