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In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group.

Attributes	Values
rdf:type	yago:WikicatSubgroupProperties yago:Abstraction100002137 yago:Possession100032613 yago:Property113244109 yago:Relation100031921 ethnic group
rdfs:label	Charakteristische Untergruppe (de) Characteristic subgroup (en) Sous-groupe caractéristique (fr) Subgrup karakteristik (in) Sottogruppo caratteristico (it) 特性部分群 (ja) 특성 부분군 (ko) Karakteristieke ondergroep (nl) Podgrupa charakterystyczna (pl) Характеристическая подгруппа (ru) Характеристична підгрупа (uk)
rdfs:comment	In der Gruppentheorie ist eine charakteristische Untergruppe einer Gruppe eine Untergruppe , die unter jedem Automorphismus von in sich abgebildet wird. (de) In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group. (en) Dans un groupe G, un sous-groupe H est dit * caractéristique lorsqu'il est stable par tout automorphisme de G : * strictement caractéristique lorsqu'il est même stable par tout endomorphisme surjectif de G ; * pleinement caractéristique, ou encore pleinement invariant, lorsqu'il est même stable par tout endomorphisme de G : (fr) Dalam matematika, khususnya di bidang aljabar abstrak yang dikenal sebagai teori grup, subgrup karakteristik adalah subgrup yang dipetakan ke dirinya sendiri oleh setiap dari induk grup. Karena setiap adalah , setiap subgrup karakteristik adalah normal; meskipun kebalikannya tidak dijamin. Contoh subgrup karakteristik termasuk subgrup komutator dan . (in) In matematica, un sottogruppo caratteristico di un gruppo è un sottogruppo tale che per ogni automorfismo di . Esempi di sottogruppi caratteristici sono il sottogruppo banale formato dal solo elemento neutro di , stesso, il centro e il sottogruppo derivato di . (it) 군론에서 특성 부분군(特性部分群, 영어: characteristic subgroup)은 주어진 군의 모든 자기 동형 사상에 대하여 불변인 부분군이다. (ko) In de groepentheorie, een deelgebied van de wiskunde, is een karakteristieke ondergroep een ondergroep die gesloten is onder alle automorfismen van de groep zelf. Deze eis is sterker dan voor een normale ondergroep, en zwakker dan het vereiste voor de . Voorbeelden van karakteristieke ondergroepen zijn de en het centrum van een groep. (nl) 数学、とくに群論という抽象代数学の分野において、特性部分群 (英: characteristic subgroup) はもとの群のすべての自己同型写像の下でな部分群である。共役は自己同型であるから、すべての特性部分群は正規部分群であるが、すべての正規部分群が特性部分群であるわけではない。特性部分群の例には、交換子部分群や群の中心がある。 (ja) Podgrupa charakterystyczna – podgrupa niezmiennicza ze względu na działanie automorfizmów. (pl) Характеристическая подгруппа — подгруппа, инвариантная относительно всех автоморфизмов группы. (ru) Характеристична підгрупа — підгрупа, інваріантна відносно всіх автоморфізмів групи. Тобто підгрупа є характеристичною, якщо для кожного автоморфізму групи і для кожного елемента виконуєься . (uk)
dcterms:subject	Subgroup properties
Wikipage page ID	7198 (xsd:integer)
Wikipage revision ID	1095663965 (xsd:integer)
Link from a Wikipage to another Wikipage	Derived subgroup Characteristically simple group Index of a subgroup Invariant (mathematics) Mathematics Normal subgroup Fully characteristic subgroup Quotient group Endomorphism Conjugacy class Commutator subgroup Identity component Subgroup Automorphism Topological group Transitive relation Verbal subgroup Cyclic group Outer automorphism group Quaternion group Group (mathematics) Surjective Abelian group Abstract algebra Subgroup properties DIH Direct product of groups Free group Group isomorphism Group theory Inner automorphism Klein four-group Symmetric group Finite group Torsion subgroup Center of a group dbr:Reduced_free_group
sameAs	Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup Characteristic subgroup
dbp:wikiPageUsesTemplate	dbt:) dbt:= dbt:Anchor dbt:Main dbt:Math dbt:Reflist dbt:Short_description dbt:Sub dbt:Sup dbt:Vanchor
has abstract	In der Gruppentheorie ist eine charakteristische Untergruppe einer Gruppe eine Untergruppe , die unter jedem Automorphismus von in sich abgebildet wird. (de) In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group. (en) Dans un groupe G, un sous-groupe H est dit * caractéristique lorsqu'il est stable par tout automorphisme de G : * strictement caractéristique lorsqu'il est même stable par tout endomorphisme surjectif de G ; * pleinement caractéristique, ou encore pleinement invariant, lorsqu'il est même stable par tout endomorphisme de G : (fr) Dalam matematika, khususnya di bidang aljabar abstrak yang dikenal sebagai teori grup, subgrup karakteristik adalah subgrup yang dipetakan ke dirinya sendiri oleh setiap dari induk grup. Karena setiap adalah , setiap subgrup karakteristik adalah normal; meskipun kebalikannya tidak dijamin. Contoh subgrup karakteristik termasuk subgrup komutator dan . (in) In matematica, un sottogruppo caratteristico di un gruppo è un sottogruppo tale che per ogni automorfismo di . Esempi di sottogruppi caratteristici sono il sottogruppo banale formato dal solo elemento neutro di , stesso, il centro e il sottogruppo derivato di . (it) 군론에서 특성 부분군(特性部分群, 영어: characteristic subgroup)은 주어진 군의 모든 자기 동형 사상에 대하여 불변인 부분군이다. (ko)

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