In mathematics, especially in the area of abstract algebra known as group theory, a verbal subgroup is any subgroup of a group definable as the subgroup generated by the set of all elements formed by choices of elements for a given set of words. For example, given the word xy, the corresponding verbal subgroup of
{
x
y
}
would be generated by the set of all products of two elements in the group, substituting any element for x and any element for y, and hence would be the group itself. On the other hand the verbal subgroup of
{
x
2
,
x
y
2
x
−
1
}
would be generated by the set of squares and their conjugates. Verbal subgroups are particularly important as the only fully characteristic subgroups of a free group and therefore represent the generic example of fully characteristic subgroups, (Magnus, Karrass & Solitar 2004, p. 75).
Another example is the verbal subgroup of
x
−
1
y
−
1
x
y
, which is the derived subgroup.
