This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: poweringinvariant subgroup and normal subgroup
View other subgroup property conjunctions  view all subgroup properties
Definition
A subgroup of a group is termed a poweringinvariant normal subgroup if it is both a poweringinvariant subgroup and a normal subgroup of the whole group. Here, poweringinvariant means that for any prime number such that is powered over , we have that is also powered over .
Relation with other properties
Stronger properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions

rationally powered normal subgroup 
normal subgroup that is powered over all primes. 


FULL LIST, MORE INFO

quotientpoweringinvariant subgroup 
the quotient group is powered over any prime that the whole group is powered over. 
quotientpoweringinvariant implies poweringinvariant 
poweringinvariant and normal not implies quotientpoweringinvariant 
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finite normal subgroup 
finite and a normal subgroup 


Quotientpoweringinvariant subgroupFULL LIST, MORE INFO

normal subgroup of finite index 
normal subgroup of finite index in the whole group. 


Quotientpoweringinvariant subgroupFULL LIST, MORE INFO

normal subgroup contained in the hypercenter 



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endomorphism kernel 
normal subgroup that is the kernel of an endomorphism 
(via quotientpoweringinvariant) 
any normal subgroup of a finite group that is not an endomorphism kernel works. 
Quotientpoweringinvariant subgroupFULL LIST, MORE INFO

complemented normal subgroup 
normal subgroup with a (possibly nonnormal) complement 
[(via endomorphism kernel) 

Endomorphism kernel, Quotientpoweringinvariant subgroupFULL LIST, MORE INFO

direct factor 
normal subgroup with normal complement 
(via complemented normal) 
(via complemented normal) 
Endomorphism kernel, Quotientpoweringinvariant subgroupFULL LIST, MORE INFO

characteristic subgroup of abelian group 
characteristic subgroup and the whole group is an abelian group. 
Characteristic subgroup of abelian group is poweringinvariant and characteristic implies normal 

Characteristic subgroup of center, Poweringinvariant characteristic subgroup, Quotientpoweringinvariant characteristic subgroup, Quotientpoweringinvariant subgroupFULL LIST, MORE INFO

Weaker properties