Editing Generating set of a group (section)

{{Use American English|date = January 2019}}
{{Short description|Abstract algebra concept}}
[[File:One5Root.svg|thumb|The 5th [[roots of unity]] in the complex plane form a [[group (mathematics)|group]] under multiplication. Each non-identity element generates the group.]]
In [[abstract algebra]], a '''generating set of a group''' is a [[subset]] of the group set such that every element of the [[group (mathematics)|group]] can be expressed as a combination (under the group operation) of finitely many elements of the subset and their [[Inverse element|inverses]].

In other words, if <math>S</math> is a subset of a group <math>G</math>, then <math>\langle S\rangle</math>, the ''subgroup generated by <math>S</math>'', is the smallest [[subgroup]] of <math>G</math> containing every element of <math>S</math>, which is equal to the intersection over all subgroups containing the elements of <math>S</math>; equivalently, <math>\langle S\rangle</math> is the subgroup of all elements of <math>G</math> that can be expressed as the finite product of elements in <math>S</math> and their inverses. (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.)

If <math>G=\langle S\rangle</math>, then we say that <math>S</math> ''generates'' <math>G</math>, and the elements in <math>S</math> are called ''generators'' or ''group generators''. If <math>S</math> is the empty set, then <math>\langle S\rangle</math> is the [[trivial group]] <math>\{e\}</math>, since we consider the [[empty product]] to be the identity.

When there is only a single element <math>x</math> in <math>S</math>, <math>\langle S\rangle</math> is usually written as <math>\langle x\rangle</math>. In this case, <math>\langle x\rangle</math> is the ''cyclic subgroup'' of the powers of <math>x</math>, a [[cyclic group]], and we say this group is generated by <math>x</math>. Equivalent to saying an element <math>x</math> generates a group is saying that <math>\langle x\rangle</math> equals the entire group <math>G</math>. For [[finite group]]s, it is also equivalent to saying that <math>x</math> has [[order (group theory)|order]] <math>|G|</math>.

A group may need an infinite number of generators. For example the additive group of [[rational number]]s <math>\Q</math> is not finitely generated. It is generated by the inverses of all the integers, but any finite number of these generators can be removed from the generating set without it ceasing to be a generating set. In a case like this, all the elements in a generating set are nevertheless "non-generating elements", as are in fact all the elements of the whole group&nbsp;− see [[Frattini subgroup]] below.

If <math>G</math> is a [[topological group]] then a subset <math>S</math> of <math>G</math> is called a set of ''topological generators'' if <math>\langle S\rangle</math> is [[Dense set|dense]] in <math>G</math>, i.e. the [[closure (topology)|closure]] of <math>\langle S\rangle</math> is the whole group <math>G</math>.