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* There is a surjective map from a [[free group]] on that many generators to the given group, that sends the generators of the free group to the elements of this ''generating set''.

<section begin=beginner/>

The elements of the generating set are termed '''generators''' (the term is best used collectively for the generating set, rather than for the elements in isolation).

===Definition with symbols===

* If <math>H</math> is a [[proper subgroup]] of <math>G</math> (i.e. <math>H</math> is a [[subgroup]] of <math>G</math> that is not equal to the whole of <math>G</math>), then <math>H</math> cannot contain <math>S</math>.<section end=beginner/>

* Consider the natural map from the free group on as many generators as elements of <math>S</math>, to the group <math>G</math>, which maps the freely generating set to the elements of <math>S</math>. This gives a surjective homomorphism from the free group, to <math>G</math>.

===Equivalence of definitions===

{{further|[[Equivalence of definitions of generating set]]}}

==Constructs==

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