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 $\genby{X}$ , 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 $|G|$ .