Verbal subgroup

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 ${\displaystyle \{xy\}}$ 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 ${\displaystyle \{x^{2},xy^{2}x^{-1}\}}$ 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 ${\displaystyle x^{-1}y^{-1}xy}$, which is the derived subgroup.

References

• Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (2004), Combinatorial Group Theory, New York: Dover Publications, ISBN 978-0-486-43830-6, MR 0207802