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Direct factor

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Definition

QUICK PHRASES: factor in internal direct product, normal with normal complement, has centralizing complement

Definition in tabular form

A direct factor of a group is defined in the following equivalent ways:

No. Shorthand A subgroup of a group is a direct factor if ... A subgroup H of a group G is a direct factor of G if ...
1 factor in internal direct product its internal direct product with some subgroup is the whole group there is a subgroup K of G such that G is the internal direct product of H and K
2 normal with normal complement it is a normal subgroup with a normal complement, i.e., it is both a normal subgroup and a retract H is normal and there is a normal subgroup K of G such that the product HK = G and H \cap K is trivial.
3 has centralizing complement there is a subgroup centralizing it, intersecting it trivially, and whose product with it is the whole group there is a subgroup K of G such that K \le C_G(H) (where CG(H) is the centralizer in G of H), H \cap K is trivial, and HK = G.

Equivalence of definitions

The equivalence of definitions follows largely from the equivalence of internal and external direct product.

Contents

This article is about a standard (though not very rudimentary) definition in group theory.[SHOW MORE]
This article defines a subgroup property that is pivotal (viz important) among existing subgroup properties
View a list of pivotal subgroup properties | View a complete list of subgroup properties[SHOW MORE]
This is a variation of normal subgroup
Find other variations of normal subgroup | Read a survey article on varying normal subgroup
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and retract
View other subgroup property conjunctions | view all subgroup properties

Examples

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Extreme examples

Every group is the internal direct product of itself and the trivial subgroup. Thus:

  1. The trivial subgroup is a direct factor of the whole group.
  2. Every group is a direct factor of itself.

High occurrence examples

  1. In a finite nilpotent group, all the Sylow subgroups are direct factors. In particular, a finite nilpotent group is the direct product of its Sylow subgroups. Further information: equivalence of definitions of finite nilpotent group
  2. In a vector space, any vector subspace is a direct factor, because the complementary subspace can be taken as the complement for an internal direct product.

Relationship with external direct product and restricted external direct product

  1. If a group G arises as the external direct product of finitely or infinitely many groups G_i, i \in I, then for any subset J \subseteq I, the subset of G arising as those elements where all coordinates outside of J are trivial is a direct factor of G. The complementary factor can be taken as the subgroup of G where all coordinates in J are trivial.
  2. A similar observation holds for the restricted external direct product.

Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Monadic second-order description

This subgroup property is a monadic second-order subgroup property, viz., it has a monadic second-order description in the theory of groups
View other monadic second-order subgroup properties

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Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Fully invariant direct factor direct factor and a fully invariant subgroup click here
Characteristic direct factor direct factor and a characteristic subgroup
Abelian direct factor direct factor and an abelian group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Central factor product with centralizer is whole group direct factor implies central factor central factor not implies direct factor click here
Complemented normal subgroup normal subgroup with a (not necessarily normal) complement complemented normal not implies direct factor click here
Retract subgroup with a normal complement direct factor implies retract retract not implies direct factor
Normal subgroup invariant under all inner automorphisms direct factor implies normal normal not implies direct factor click here
Permutably complemented subgroup there exists a permutable complement: a subgroup intersecting it trivially and such that their product is the whole group click here
Lattice-complemented subgroup there exists a lattice complement: a subgroup whose intersection with it is trivial and join with it is the whole group click here
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Related group properties

Group property Definition in terms of direct factor
Directly indecomposable group nontrivial group with no proper nontrivial direct factor
Complete group it is a direct factor of any bigger group in which it is a normal subgroup
Group in which every normal subgroup is a direct factor every normal subgroup is a direct factor

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
Transitive subgroup property Yes direct factor is transitive If M \le H \le G, with M a direct factor of H and H a direct factor of G, then M is a direct factor of G.
Finite-intersection-closed subgroup property No direct factor is not finite-intersection-closed We can have H,M direct factors of G but H \cap M not a direct factor of G.
Finite-join-closed subgroup property No direct factor is not finite-join-closed We can have H,M direct factors of G but \langle H, M \rangle not a direct factor of G.
Intermediate subgroup condition Yes direct factor satisfies intermediate subgroup condition If H \le M \le G with H a direct factor of G, then H is a direct factor of M.
Trim subgroup property Yes The whole group and the trivial subgroup are direct factors
Image condition No direct factor does not satisfy image condition We can have a surjective homomorphism \varphi:G \to L and a direct factor H of G such that \varphi(H) is not a direct factor of L.
Quotient-transitive subgroup property Yes direct factor is quotient-transitive If H \le M \le G with H a direct factor of G and M / H a direct factor of G / H, then M is a direct factor of G.
upper join-closed subgroup property No direct factor is not upper join-closed We can have H \le G and M1,M2 intermediate subgroups such that H is a direct factor in each but not in \langle M_1, M_2 \rangle.
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Effect of property operators

The finite-join-closure

Applying the finite-join-closure to this property gives: join of finitely many direct factors

The join-closure

Applying the join-closure to this property gives: join of direct factors

The image-potentially operator

Applying the image-potentially operator to this property gives: central factor

A subgroup H of a group G is a central factor if and only if there exists a surjective homomorphism of groups \rho:K \to G such that ρ − 1(H) is a direct factor of K. For full proof, refer: Central factor iff image-potentially direct factor

Testing

GAP code

One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup property at: IsDirectFactor
View other GAP-codable subgroup properties | View subgroup properties with in-built commands
Facts about Direct factorRDF feed
Applying operator givesJoin of finitely many direct factors  +, Join of direct factors  +, and Central factor  +
Conjunction involvingNormal subgroup  +, and Retract  +
Defining ingredientInternal direct product  +, Normal subgroup  +, Normal complement  +, and Retract  +
Dissatisfies metapropertyIntersection-closed subgroup property  +, Strongly intersection-closed subgroup property  +, Join-closed subgroup property  +, Finite-intersection-closed subgroup property  +, Finite-join-closed subgroup property  +, and Image condition  +
Dissatisfies propertyUpper join-closed subgroup property  +
Page classTerm  +
Quick phrasefactor in internal direct product  +, normal with normal complement  +, and has centralizing complement  +
Satisfies metapropertyTransitive subgroup property  +, Intermediate subgroup condition  +, Trim subgroup property  +, Trivially true subgroup property  +, Identity-true subgroup property  +, Left-realized subgroup property  +, Right-realized subgroup property  +, Quotient-transitive subgroup property  +, and Monadic second-order subgroup property  +
Stronger thanRight-quotient-transitively central factor  +, Complemented central factor  +, Join-transitively central factor  +, Conjugacy-closed subgroup  +, AEP-subgroup  +, Normal AEP-subgroup  +, Intermediately AEP-subgroup  +, Normal intermediately AEP-subgroup  +, Subgroup in which every subgroup characteristic in the whole group is characteristic  +, Normal subgroup in which every subgroup characteristic in the whole group is characteristic  +, Intersection of direct factors  +, Direct factor of characteristic subgroup  +, Base of a wreath product  +, Transitively normal subgroup  +, Conjugacy-closed normal subgroup  +, SCAB-subgroup  +, Central factor  +, Complemented normal subgroup  +, Retract  +, Normal subgroup  +, Permutably complemented subgroup  +, and Lattice-complemented subgroup  +
Variation ofNormal subgroup  +
Weaker thanFully invariant direct factor  +, Characteristic direct factor  +, and Abelian direct factor  +
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