Fully characteristic subgroup

Fully characteristic subgroup

In mathematics, a subgroup of a group is fully characteristic (or fully invariant) 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.

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.

The commutator subgroup of a group is always a fully characteristic subgroup. More generally, any verbal subgroup is always fully characteristic. For any reduced free group, and, in particular, for any free group, the converse also holds — every fully characteristic subgroup is verbal.

See also characteristic subgroup.

References

*cite book | title = Group Theory | first = W.R. | last = Scott | pages = 45-46 | publisher = Dover | year = 1987 | id = ISBN 0-486-65377-3
*cite book | title = Combinatorial Group Theory | first = Wilhelm | last = Magnus | coauthors = Abraham Karrass, Donald Solitar | publisher = Dover | year = 2004 | pages = 74-85 | id = ISBN 0-486-43830-9


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • 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.[1][2] Because conjugation is an automorphism, every… …   Wikipedia

  • Normal subgroup — Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum semidirect product, wreath product …   Wikipedia

  • Torsion subgroup — In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order. An abelian group A is called a torsion (or periodic) group if every element of A has finite… …   Wikipedia

  • 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.[1][2] The commutator subgroup is important because it is the smallest… …   Wikipedia

  • List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… …   Wikipedia

  • Charakteristische Untergruppe — In der Gruppentheorie ist eine charakteristische Untergruppe einer Gruppe G eine Untergruppe H, die unter jedem Automorphismus von G fest bleibt. Das heißt, eine Untergruppe H von G heißt charakteristisch, wenn für jeden Automorphismus… …   Deutsch Wikipedia

  • Center (group theory) — In abstract algebra, the center of a group G is the set Z ( G ) of all elements in G which commute with all the elements of G . That is,:Z(G) = {z in G | gz = zg ;forall,g in G}.Note that Z ( G ) is a subgroup of G , because # Z ( G ) contains e …   Wikipedia

  • Modular representation theory — is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic. As well as having applications to group theory, modular representations arise… …   Wikipedia

  • Nigeria — Nigerian, adj., n. /nuy jear ee euh/, n. a republic in W Africa: member of the Commonwealth of Nations; formerly a British colony and protectorate. 107,129,469; 356,669 sq. mi. (923,773 sq. km). Cap.: Abuja. Official name, Federal Republic of… …   Universalium

  • arts, East Asian — Introduction       music and visual and performing arts of China, Korea, and Japan. The literatures of these countries are covered in the articles Chinese literature, Korean literature, and Japanese literature.       Some studies of East Asia… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”