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Generating set of a group

204 bytes added, 15:12, 31 July 2008
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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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