Generating set of a group

In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.

, which is equal to the intersection over all subgroups containing the elements of

that can be expressed as the finite product of elements in

(Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.)

A group may need an infinite number of generators.

For example the additive group of rational numbers

is called a set of topological generators if

The structure of finitely generated abelian groups in particular is easily described.

Many theorems that are true for finitely generated groups fail for groups in general.

It has been proven that if a finite group is generated by a subset

, then each group element may be expressed as a word from the alphabet

of length less than or equal to the order of the group.

The integers under addition are an example of an infinite group which is finitely generated by both 1 and −1, but the group of rationals under addition cannot be finitely generated.

No uncountable group can be finitely generated.

For example, the group of real numbers under addition,

also generates the group of integers under addition by Bézout's identity.

is isomorphic to the free group in countably infinitely many generators, and so cannot be finitely generated.

In fact, more can be said: the class of all finitely generated groups is closed under extensions.

The most general group generated by a set

An interesting companion topic is that of non-generators.

The set of all non-generators forms a subgroup of

is a semigroup or a monoid, one can still use the notion of a generating set

The definitions of generating set of a group using finite sums, given above, must be slightly modified when one deals with semigroups or monoids.

Indeed, this definition should not use the notion of inverse operation anymore.

For example, {1} is a monoid generator of the set of natural numbers

The set {1} is also a semigroup generator of the positive natural numbers

However, the integer 0 can not be expressed as a (non-empty) sum of 1s, thus {1} is not a semigroup generator of the natural numbers.

Similarly, while {1} is a group generator of the set of integers

, {1} is not a monoid generator of the set of integers.

Indeed, the integer −1 cannot be expressed as a finite sum of 1s.