Tour:Generating set of a group

This article adapts material from the main article: generating set of a group

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Definition

Symbol-free definition

A subset of a group is termed a generating set if it satisfies the following equivalent conditions:

• Every element of the group can be expressed in terms of the elements of this subset by means of the group operations of multiplication and inversion.
• There is no proper subgroup of the group containing this subset
• There is a surjective map from a free group on that many generators to the given group, that sends the generators of the free group to the elements of this generating set.

The elements of the generating set are termed generators.

Definition with symbols

A subset $S$ of a group $G$ is termed a generating set if it satisfies the following equivalent conditions:

• For any element $g\in G$, we can write:

$g=a_1{a}_2\dots a_n$

where for each $a_i$, either $a_i\in S$ or $a_i^{-1}\in S$ (here, the $a_i$s are not necessarily distinct).

• If $H$ is a proper subgroup of $G$ (i.e. $H$ is a subgroup of $G$ that is not equal to the whole of $G$), then $H$ cannot contain $S$.
• Consider the natural map from the free group on as many generators as elements of $S$, to the group $G$, which maps the freely generating set to the elements of $S$. This gives a surjective homomorphism from the free group, to $G$.