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Generating set of a group

48 bytes added, 04:37, 25 August 2007
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.
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]].
 
==Study of this notion==
 
{{msc class|20F05}}
[[Category: Properties of subsets of groups]]
[[Category: Terminology related to combinatorial group theory]]
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