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Generating set of a group

1,666 bytes added, 15:08, 21 February 2007
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#redirect ==Definition== ===Symbol-free definition=== A [[Generating subset of a group]] is termed a '''generating set''' if it satisfies the following equivalent conditions: * Every element of the group can be expressed in terms of the elements of this subset by means of the group operations of multiplication and inversion.* There is no proper subgroup of the group containing this subset* 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''. The elements of the generating set are termed generators. ===Definition with symbols=== {{fillin}} ==Constructs== ===Cayley graph=== {further|[[Cayley graph of a group]]}}Given a generating set of a group, we can construct the [[Cayley graph of a group|Cayley graph]] of the group with respect to that generating set. This is a graph whose vertex set is the set of elements of the group and where there is an edge between two vertices whenever one can be taken to the other by left multiplying by a generator. ===Presentation=== The generators of a group may not in general be independent. That is, there may be words in the generators that simplify to the identity in the given group. Such a word is termed a [[relation]] between the generators. Relations can be viewed as elements of the free group on symbols corresponding to the generators. Thus relations form a [[normal subgroup]] of the free group. Specifying a set of relations whose normal closure is this normal subgroup is what we call a [[presentation of a group]]. [[Category: Properties of subsets of groups]][[Category: Terminology related to combinatorial grouptheory]]
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