Generator: Difference between revisions

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A '''generating set''' of a [[Wikipedia: Group (mathematics)|group]] is a subset of the elements of the group which is not contained in any [[Wikipedia: 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 [[Wikipedia: Abelian group|abelian group]], it is called a [[Wikipedia: Finitely generated abelian group|finitely generated abelian group]]. An element of a generating set is called a generator.

A '''minimal generating set''' is a generating set which has no "redundant" or "unnecessary" generators. In [[Wikipedia: Free abelian group|free abelian groups]] such as [[just intonation subgroup]]s or its [[regular temperament]]s, this is the same thing as a [[basis]]. 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.

A '''minimal generating set''' is a generating set which has no "redundant" or "unnecessary" generators. In [[Wikipedia: Free abelian group|free abelian groups]] such as [[just intonation subgroup]]s or its [[regular temperament]]s, this is the same thing as a [[basis]]. 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 and the set will remain a generating set.

If the group operation is written additively, then if <math>\lbrace g_1, g_2, \ldots g_k \rbrace</math> is the generating set, every element <math>g</math> of the group can be written

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=== Convention ===

In [[rank|multirank]] systems, it is customary that generators are said as opposed to the period. Specifically, the first generator is called the period, and only the rest are called the generators.

== See also ==

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 A '''generating set''' of a [[Wikipedia: Group (mathematics)|group]] is a subset of the elements of the group which is not contained in any [[Wikipedia: 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 [[Wikipedia: Abelian group|abelian group]], it is called a [[Wikipedia: Finitely generated abelian group|finitely generated abelian group]]. An element of a generating set is called a generator.
-A '''minimal generating set''' is a generating set which has no "redundant" or "unnecessary" generators. In [[Wikipedia: Free abelian group|free abelian groups]] such as [[just intonation subgroup]]s or its [[regular temperament]]s, this is the same thing as a [[basis]]. 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.
+A '''minimal generating set''' is a generating set which has no "redundant" or "unnecessary" generators. In [[Wikipedia: Free abelian group|free abelian groups]] such as [[just intonation subgroup]]s or its [[regular temperament]]s, this is the same thing as a [[basis]]. 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 and the set will remain a generating set.
 If the group operation is written additively, then if <math>\lbrace g_1, g_2, \ldots g_k \rbrace</math> is the generating set, every element <math>g</math> of the group can be written
@@ Line 25: / Line 25: @@
 === Convention ===
 In [[rank|multirank]] systems, it is customary that generators are said as opposed to the period. Specifically, the first generator is called the period, and only the rest are called the generators.
 == See also ==