In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the area of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
known as
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a verbal subgroup is a
subgroup of a
group that is
generated by all elements that can be formed by substituting group elements for variables in a given set of
words.
For example, given the word ''xy'', the corresponding verbal subgroup is 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 for the set of words
is generated by the set of squares and their
conjugates. Verbal subgroups are the only
fully characteristic subgroups of a
free group and therefore represent the generic example of fully characteristic subgroups, .
Another example is the verbal subgroup for
, which is the
derived subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
.
References
*
Infinite group theory
Subgroup properties
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