Minimum size of generating set: Difference between revisions

Revision as of 20:27, 31 July 2009

This article defines an arithmetic function on groups
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Definition

Let $G$ be a group. The minimum size of generating set for $G$ , often called the rank or generating set-rank of $G$ , and sometimes denoted $d(G)$ or $r(G)$ , is defined as the minimum possible size of a generating set for $G$ .

This number is finite if and only if the group is a finitely generated group.

Related notions

Subgroup rank of a group: This is the maximum of the generating set-ranks over all subgroups of the group.
Rank of a p-group: For a group of prime power order, this is the maximum of the ranks of all the abelian subgroups of the group.

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 ==Definition==
-Let <math>G</math> be a [[finitely generated group]]. The '''minimum size of generating set''' for <math>G</math>, often called the '''rank''' or '''generating set-rank''' of <math>G</math>, and sometimes denoted <math>d(G)</math> or <math>r(G)</math>, is defined as the minimum possible size of a [[defining ingredient::generating set of a group|generating set]] for <math>G</math>.
+Let <math>G</math> be a [[group]]. The '''minimum size of generating set''' for <math>G</math>, often called the '''rank''' or '''generating set-rank''' of <math>G</math>, and sometimes denoted <math>d(G)</math> or <math>r(G)</math>, is defined as the minimum possible size of a [[defining ingredient::generating set of a group|generating set]] for <math>G</math>.
+This number is finite if and only if the group is a [[finitely generated group]].
 ==Related notions==