In mathematics

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...

, a 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...

 is fully characteristic (or fully invariant) if it is invariant

Invariant (mathematics)

In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...

 under every endomorphism

Endomorphism

In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...

 of the group. That is, any endomorphism of the group takes elements of the subgroup to elements of the subgroup.

Every group has itself (the improper subgroup) and the trivial subgroup as two of its fully characteristic subgroups. Every fully characteristic subgroup is a strictly characteristic subgroup, and a fortiori a characteristic subgroup

Characteristic subgroup

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group. Because conjugation is an automorphism, every characteristic subgroup is normal, though not every normal...

.

The commutator subgroup

Commutator 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....

 of a group is always a fully characteristic subgroup. More generally, any verbal subgroup

Verbal subgroup

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...

 is always fully characteristic. For any reduced free group, and, in particular, for any 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...

, the converse also holds — every fully characteristic subgroup is verbal.

See also characteristic subgroup

Characteristic subgroup

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group. Because conjugation is an automorphism, every characteristic subgroup is normal, though not every normal...

.