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Generating set of a group

871 bytes added, 21:33, 22 March 2008
===Definition with symbols===
A subset <math>S</math> of a group <math>G</math> is termed a '''generating set''' if it satisfies the following equivalent conditions: * For any element <math>g \in G</math>, we can write: <math>g = a_1a_2 \ldots a_n</math> where for each <math>a_i</math>, either <math>a_i \in S</math> or <math>a_i^{{fillin}-1}\in S</math> (here, the <math>a_i</math>s are not necessarily distinct).* 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>.* 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>.
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