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Generating set of a group

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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 <math>S</math> is a subset of a group <math>G</math> such that every element of <math>G</math> is a power of some element of <math>S</math>, then <math>S</math> is a generating set for <math>G</math>.
 
===Some examples in Abelian groups===
 
* In the group of integers under addition, the singleton set <math>\{ 1 \}</math> is a generating set. This is because every integer can be written as a sum of <math>1</math>s or <math>-1</math>s.
* In the group of integers under addition, the two-element set <math>\{ 2,3 \}</math> is a generating set. To see this, note that every integer can expressed as a sum of <math>1</math>s or <math>-1</math>s, and both <math>1</math> and <math>-1</math> can be expressed in terms of <math>2</math> and <math>3</math>.
* In the group of rational numbers, the set of all unit fractions <math>1/n, n \in \mathbb{N}</math>, 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 <math>SL(n,k)</math> over a field <math>k</math> is generated by elementary matrices. {{proofat|[[Elementary matrices generate the special linear group]]}}
 
For more information on generating sets of particular groups, refer:
 
[[:Category:Generating sets for particular groups]]
==Constructs==
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