# Changes

## Generating set of a group

, 16:28, 8 December 2008
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===Equivalence of definitions===
{{further|[[Equivalence of definitions of generating set]]}}
<section begin=beginner/>
==Examples==
===Extreme examples===

* The set of ''all'' elements of a group is a generating set for the group. It is also the largest possible generating set.
* The set of all non-identity elements of a group is a generating set for the group.
* If $S$ is a subset of a group $G$ such that every element of $G$ is a power of some element of $S$, then $S$ is a generating set for $G$.

===Some examples in Abelian groups===

* In the group of integers under addition, the singleton set $\{ 1 \}$ is a generating set. This is because every integer can be written as a sum of $1$s or $-1$s.
* In the group of integers under addition, the two-element set $\{ 2,3 \}$ is a generating set. To see this, note that every integer can expressed as a sum of $1$s or $-1$s, and both $1$ and $-1$ can be expressed in terms of $2$ and $3$.
* In the group of rational numbers, the set of all unit fractions $1/n, n \in \mathbb{N}$, form a generating set.
<section end=beginner/>

===Some examples in non-Abelian groups===

* In the symmetric group on a finite set, the set of all transpositions is a generating set for the group. {{proofat|[[Transpositions generate the finitary symmetric group]]}}
* The special linear group $SL(n,k)$ over a field $k$ is generated by elementary matrices. {{proofat|[[Elementary matrices generate the special linear group]]}}