Verbal subgroup
Encyclopedia
In mathematics
, especially in the area of abstract algebra
known as group theory
, a verbal subgroup is a special sort of subgroup
of a group
that can be defined as all elements that can be expressed as products of a special form, called a word. Verbal subgroups are particularly important as the only fully characteristic subgroup
s of a free group
and therefore represent the generic example of fully characteristic subgroups, .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the area of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
known as group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a verbal subgroup is a special sort of subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
that can be defined as all elements that can be expressed as products of a special form, called a word. Verbal subgroups are particularly important as the only fully characteristic subgroup
Fully characteristic subgroup
In mathematics, a subgroup of a group is fully characteristic if it is invariant under every endomorphism of the group. That is, any endomorphism of the group takes elements of the subgroup to elements of the subgroup....
s of a free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
and therefore represent the generic example of fully characteristic subgroups, .