A subset X of a group G is a generating set if every element of G can be written as a finite product of elements from X or their inverses.
We have already mentioned this for cyclic groups where a generating set
need only contain one element. Indeed, that characterizes cyclic groups.
A cyclic group of order 6 with generator x can also be generated by
since 
, 
, 
.
On a similar note, we define 
to be the set of all finite
products of elements of X or their inverses. In fact,
is a subgroup of G and is called, not surprisingly, the subgroup generated
by X. Clearly, X generates G if and only if 
.
As another example, the symmetry group of the square 
is generated by
the rotation through 
and the reflection across the vertical.
It is also generated by the reflection about the vertical and the reflection
about the horizontal.
The symmetric group on n letters, 
is generated by the transpositions.
The size of this generating set is 
.
Another (smaller) generating set consists of (1 2) and 
.
We say that G is finitely generated if it has a finite generating set. The integers for example is finitely generated whereas the rational numbers under addition is not finitely generated.
By convention, if X is the empty set then the (sub)group it generates is the trivial group.
One misconception about generating sets is that if 
generate a
group G (so that 
) it is not the case that G is
made up only of the powers of a with the powers of b. That is,
. Consider,
as an example, the group 
. It is generated by 
and
. The powers of these elements would account for only 5 of the
24 elements of . Indeed, even products of the form 
could
account for at most eight elements of the group.