Vipul: /* Weaker properties */

2013-03-01T17:36:01Z

Weaker properties

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Vipul: Created page with "{{in-group subgroup property|powering-invariant subgroup|abelian group}} ==Definition== Suppose $G$ is an group and $H$ is a subgroup of $G$ ..."

2013-03-01T17:35:41Z

Created page with "{{in-group subgroup property|powering-invariant subgroup|abelian group}} ==Definition== Suppose <math>G</math> is an group and <math>H</math> is a subgroup of <math>G</math>..."

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{{in-group subgroup property|powering-invariant subgroup|abelian group}}

==Definition==

Suppose <math>G</math> is an group and <math>H</math> is a subgroup of <math>G</math>. We say that <math>H</math> in <math>G</math> is a '''powering-invariant subgroup of abelian group''' if the following equivalent conditions are satisfied:

# <math>G</math> is an [[abelian group]] and <math>H</math> is a [[powering-invariant subgroup]] of <math>G</math>.
# <math>G</math> is an [[abelian group]] and <math>H</math> is a [[quotient-powering-invariant subgroup]] of <math>G</math>.

==Examples==

===Non-examples===

[[Group of integers in group of rational numbers]] is the standard non-example.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::characteristic subgroup of abelian group]] || the whole group is [[abelian group|abelian]] and the subgroup is [[characteristic subgroup|characteristic]]. || [[characteristic subgroup of abelian group is powering-invariant]] || any non-characteristic subgroup of a finite abelian group. || {{intermediate notions short|powering-invariant subgroup of abelian group|characteristic subgroup of abelian group}}
|-
| [[Weaker than::direct factor of abelian group]] || || || ||
|-
| [[Weaker than::subgroup of finite abelian group]] || || || ||
|}

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::powering-invariant normal subgroup of nilpotent group]] || || follows from [[abelian implies nilpotent]] and [[abelian implies every subgroup is normal]] || follows from [[nilpotent not implies abelian]] || {{intermediate notions short|powering-invariant normal subgroup of nilpotent group|powering-invariant subgroup of abelian group}}
|-
| [[Stronger than::powering-invariant subgroup]] || || || ||
|}
|}

Powering-invariant subgroup of abelian group - Revision history

Vipul: /* Weaker properties */

Vipul: Created page with "{{in-group subgroup property|powering-invariant subgroup|abelian group}} ==Definition== Suppose G is an group and H is a subgroup of G..."

Vipul: Created page with "{{in-group subgroup property|powering-invariant subgroup|abelian group}} ==Definition== Suppose $G$ is an group and $H$ is a subgroup of $G$ ..."