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generating set of a group (Definition)

Let $ G$ be a group.

A subset $ X\subseteq G$ is said to generate $ G$ (or to be a generating set of $ G$) if no proper subgroup of $ G$ contains $ X$.

A subset $ X\subseteq G$ generates $ G$ if and only if every element of $ G$ can be expressed as a product of elements of $ X$ and inverses of elements of $ X$ (taking the empty product to be the identity element). A subset $ X\subseteq G$ is said to be closed under inverses if $ x^{-1}\in X$ whenever $ x\in X$; if a generating set $ X$ of $ G$ is closed under inverses, then every element of $ G$ is a product of elements of $ X$.

A group that has a generating set with only one element is called a cyclic group. A group that has a generating set with only finitely many elements is called a finitely generated group.

If $ X$ is an arbitrary subset of $ G$, then the subgroup of $ G$ generated by $ X$, denoted by $ {\left\langle X\right\rangle}$, is the smallest subgroup of $ G$ that contains $ X$.

The generating rank of $ G$ is the minimum cardinality of a generating set of $ G$. (This is sometimes just called the rank of $ G$, but this can cause confusion with other meanings of the term rank.) If $ G$ is uncountable, then its generating rank is simply $ \vert G\vert$.



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See Also: presentation of a group, generator

Other names:  generating set
Also defines:  generate, generates, generated by, subgroup generated by, generating rank, closed under inverses, group generated by
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Cross-references: uncountable, cardinality, subgroup, finitely generated group, cyclic group, identity element, empty product, inverses, product, contains, proper subgroup, subset, group
There are 337 references to this entry.

This is version 4 of generating set of a group, born on 2005-12-30, modified 2006-08-14.
Object id is 7545, canonical name is GeneratingSetOfAGroup.
Accessed 12773 times total.

Classification:
AMS MSC: 20F05 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Generators, relations, and presentations)
 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
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