Generator: Difference between revisions

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== Generators in math and JI subgroups ==

A [http://en.wikipedia.org/wiki/Generating_set_of_a_group set of '''generators'''], or '''generating set''', for a [http://en.wikipedia.org/wiki/Group_%28mathematics%29 group] is a subset of the elements of the group which is not contained in any [http://en.wikipedia.org/wiki/Subgroup proper subgroup], which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an [http://en.wikipedia.org/wiki/Abelian_group abelian group], it is called a [http://en.wikipedia.org/wiki/Finitely_generated_abelian_group finitely generated abelian group]. A '''basis''' is a minimal generating set, i.e. a generating set which has no "redundant" or "unnecessary" generators.

A [http://en.wikipedia.org/wiki/Generating_set_of_a_group set of '''generators'''], or '''generating set''', for a [http://en.wikipedia.org/wiki/Group_%28mathematics%29 group] is a subset of the elements of the group which is not contained in any [http://en.wikipedia.org/wiki/Subgroup proper subgroup], which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an [http://en.wikipedia.org/wiki/Abelian_group abelian group], it is called a [http://en.wikipedia.org/wiki/Finitely_generated_abelian_group finitely generated abelian group].

A '''basis''' (plural ''bases'') is a minimal generating set, i.e. a generating set which has no "redundant" or "unnecessary" generators. For example, {2, 3, 5} and {2.3.5/3} are bases for the JI subgroup 2.3.5. However, {2, 3, 5, 15} is not a basis: 15 = 3*5 so we can take out 15 from this generating set.

If the abelian group is written additively, then if {g1, g2, ... gk} is the generating set, every element g of the group can be written

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 == Generators in math and JI subgroups ==
-A [http://en.wikipedia.org/wiki/Generating_set_of_a_group set of '''generators'''], or '''generating set''', for a [http://en.wikipedia.org/wiki/Group_%28mathematics%29 group] is a subset of the elements of the group which is not contained in any [http://en.wikipedia.org/wiki/Subgroup proper subgroup], which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an [http://en.wikipedia.org/wiki/Abelian_group abelian group], it is called a [http://en.wikipedia.org/wiki/Finitely_generated_abelian_group finitely generated abelian group]. A '''basis''' is a minimal generating set, i.e. a generating set which has no "redundant" or "unnecessary" generators.
+A [http://en.wikipedia.org/wiki/Generating_set_of_a_group set of '''generators'''], or '''generating set''', for a [http://en.wikipedia.org/wiki/Group_%28mathematics%29 group] is a subset of the elements of the group which is not contained in any [http://en.wikipedia.org/wiki/Subgroup proper subgroup], which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an [http://en.wikipedia.org/wiki/Abelian_group abelian group], it is called a [http://en.wikipedia.org/wiki/Finitely_generated_abelian_group finitely generated abelian group].
+A '''basis''' (plural ''bases'') is a minimal generating set, i.e. a generating set which has no "redundant" or "unnecessary" generators. For example, {2, 3, 5} and {2.3.5/3} are bases for the JI subgroup 2.3.5. However, {2, 3, 5, 15} is not a basis: 15 = 3*5 so we can take out 15 from this generating set.
 If the abelian group is written additively, then if {g1, g2, ... gk} is the generating set, every element g of the group can be written