No. |
Shorthand |
A subgroup of a group is a direct factor if ... |
A subgroup of a group is a direct factor of if ...
|
1 |
factor in internal direct product |
its internal direct product with some subgroup is the whole group |
there is a subgroup of such that is the internal direct product of and
|
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 |
is normal and there is a normal subgroup of such that the product and 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 of such that (where is the centralizer in of ), is trivial, and .
|
4 |
factor in internal direct product of multiple subgroups |
it is one of the subgroups occurring in an internal direct product decomposition of the whole group into (possibly more than two) subgroups. Note that we also allow infinitely many subgroups, in which case, the internal direct product would correspond to the restricted external direct product. |
there is a collection of subgroups with equal to one of the s, such that is the internal direct product of the s.
|
Every group is the internal direct product of itself and the trivial subgroup. Thus:
Property |
Meaning |
Proof of implication |
Proof of strictness (reverse implication failure) |
Intermediate notions
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central factor |
product with centralizer is whole group |
direct factor implies central factor |
central factor not implies direct factor (see also list of examples) |
Complemented central factor, Join of direct factor and central subgroup, Join of finitely many direct factors, Join-transitively central factor, Right-quotient-transitively central factor, Upper join of direct factors|FULL LIST, MORE INFO
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complemented normal subgroup |
normal subgroup with a (not necessarily normal) complement |
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complemented normal not implies direct factor (see also list of examples) |
Complemented central factor, Complemented transitively normal subgroup, Right-transitively complemented normal subgroup|FULL LIST, MORE INFO
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endomorphism kernel |
kernel of an endomorphism |
(via complemented normal) |
(via complemented normal) |
Complemented normal subgroup, Intermediately endomorphism kernel|FULL LIST, MORE INFO
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retract |
subgroup with a normal complement |
direct factor implies retract |
retract not implies direct factor (see also list of examples) |
|FULL LIST, MORE INFO
|
normal subgroup |
invariant under all inner automorphisms |
direct factor implies normal |
normal not implies direct factor (see also list of examples) |
Central factor, Characteristic subgroup of direct factor, Complemented central factor, Complemented normal subgroup, Complemented transitively normal subgroup, Conjugacy-closed normal subgroup, Direct factor over central subgroup, Endomorphism kernel, Intermediately endomorphism kernel, Join of finitely many direct factors, Join-transitively central factor, Locally inner automorphism-balanced subgroup, Normal AEP-subgroup, Normal subgroup having a 1-closed transversal, Normal subgroup in which every subgroup characteristic in the whole group is characteristic, Normal subgroup whose focal subgroup equals its derived subgroup, Powering-invariant normal subgroup, Quotient-powering-invariant subgroup, Right-quotient-transitively central factor, SCAB-subgroup... further results|FULL LIST, MORE INFO
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permutably complemented subgroup |
there exists a permutable complement: a subgroup intersecting it trivially and such that their product is the whole group |
|
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Complemented central factor, Complemented normal subgroup, Retract|FULL LIST, MORE INFO
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lattice-complemented subgroup |
there exists a lattice complement: a subgroup whose intersection with it is trivial and join with it is the whole group |
|
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Complemented central factor, Complemented normal subgroup, Permutably complemented subgroup, Retract, Right-transitively lattice-complemented subgroup|FULL LIST, MORE INFO
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verbally closed subgroup |
image of subgroup under word map equals its intersection with image of whole group under word map |
(via retract) |
(via retract) |
Retract|FULL LIST, MORE INFO
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local divisibility-closed subgroup |
if an element in the subgroup has a root in the whole group, it has a root in the subgroup. |
(via retract) |
(via retract) |
Retract, Verbally closed subgroup|FULL LIST, MORE INFO
|
local powering-invariant subgroup |
if an element in the subgroup has a unique root in the whole group, that root is in the subgroup. |
(via local divisibility-closed) |
(via local divisibility-closed) |
Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Retract, Verbally closed subgroup|FULL LIST, MORE INFO
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divisibility-closed subgroup |
if every element in the subgroup has a root in the whole group, every element has a root in the subgroup. |
|
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Retract, Verbally closed subgroup|FULL LIST, MORE INFO
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powering-invariant subgroup |
if every element has a unique root in the group, every element of the subgroup has a unique root in the subgroup. |
(via local divisibility-closed) |
(via local divisibility-closed) |
Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Retract, Verbally closed subgroup|FULL LIST, MORE INFO
|
powering-invariant normal subgroup |
both powering-invariant and normal |
(follows from implications for powering-invariant and normal |
|
Endomorphism kernel, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO
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quotient-powering-invariant subgroup |
normal, and if every element has a unique root in the group, every element of the quotient group has a unique root in the quotient group. |
([[endomorphism kernel implies quotient-powering-invariant|via endomorphism kernel) |
(via endomorphism kernel) |
Endomorphism kernel|FULL LIST, MORE INFO
|
right-quotient-transitively central factor |
normal subgroup such that any subgroup containing it with the quotient a central factor of the whole quotient, is also a central factor |
direct factor implies right-quotient-transitively central factor |
right-quotient-transitively central factor not implies direct factor |
|FULL LIST, MORE INFO
|
complemented central factor |
central factor with a complement |
direct factor implies complemented central factor |
complemented central factor not implies direct factor |
|FULL LIST, MORE INFO
|
join-transitively central factor |
join with any central factor is a central factor |
direct factor implies join-transitively central factor |
join-transitively central factor not implies direct factor |
Join of direct factor and central subgroup, Join of finitely many direct factors, Right-quotient-transitively central factor|FULL LIST, MORE INFO
|
conjugacy-closed subgroup |
any two elements of the subgroup conjugate in the whole group are conjugate in the subgroup |
|
|
Base of a wreath product, Central factor, Conjugacy-closed normal subgroup, Retract, Subset-conjugacy-closed subgroup|FULL LIST, MORE INFO
|
AEP-subgroup |
every automorphism of subgroup extends to an automorphism of whole group |
Direct factor implies AEP |
AEP not implies direct factor |
Base of a wreath product, Intermediately AEP-subgroup, Normal intermediately AEP-subgroup, Sectionally AEP-subgroup|FULL LIST, MORE INFO
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normal AEP-subgroup |
both a normal subgroup and an AEP-subgroup |
|
|
Normal intermediately AEP-subgroup|FULL LIST, MORE INFO
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intermediately AEP-subgroup |
AEP-subgroup in every intermediate subgroup |
|
|
Normal intermediately AEP-subgroup|FULL LIST, MORE INFO
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normal intermediately AEP-subgroup |
normal and an intermediately AEP-subgroup |
|
|
|FULL LIST, MORE INFO
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subgroup in which every subgroup characteristic in the whole group is characteristic |
every characteristic subgroup of whole group contained in the subgorup is characteristic in the subgroup |
(via AEP) |
(via AEP) |
AEP-subgroup, Normal AEP-subgroup, Normal intermediately AEP-subgroup, Normal subgroup in which every subgroup characteristic in the whole group is characteristic|FULL LIST, MORE INFO
|
normal subgroup in which every subgroup characteristic in the whole group is characteristic |
normal subgroup such that every characteristic subgroup of the whole group contained in it is characteristic in it |
(via normal AEP) |
(via normal AEP) |
Normal AEP-subgroup|FULL LIST, MORE INFO
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intersection of direct factors |
intersection of direct factors |
|
|
|FULL LIST, MORE INFO
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direct factor of characteristic subgroup |
direct factor of characteristic subgroup |
|
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Direct factor of fully invariant subgroup|FULL LIST, MORE INFO
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base of a wreath product |
base of an internal wreath product |
|
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|FULL LIST, MORE INFO
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transitively normal subgroup |
every normal subgroup of it is normal in the whole group |
|
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Central factor, Complemented transitively normal subgroup, Conjugacy-closed normal subgroup, Join of finitely many direct factors, Join-transitively central factor, Locally inner automorphism-balanced subgroup, Right-quotient-transitively central factor, SCAB-subgroup, Upper join of direct factors|FULL LIST, MORE INFO
|
conjugacy-closed normal subgroup |
conjugacy-closed and normal |
|
|
Template:Intermediate notion short
|
SCAB-subgroup |
every subgroup-conjugating automorphism of whole group restricts to a subgroup-conjugating automorphism of subgroup |
|
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Central factor, Join of finitely many direct factors, Join-transitively central factor|FULL LIST, MORE INFO
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