Powering-invariant normal subgroup
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: powering-invariant subgroup and normal subgroup
View other subgroup property conjunctions | view all subgroup properties
Definition
A subgroup of a group is termed a powering-invariant normal subgroup if it is both a powering-invariant subgroup and a normal subgroup of the whole group. Here, powering-invariant 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 | ||
| quotient-powering-invariant subgroup | the quotient group is powered over any prime that the whole group is powered over. | quotient-powering-invariant implies powering-invariant | powering-invariant and normal not implies quotient-powering-invariant | |FULL LIST, MORE INFO |
| finite normal subgroup | finite and a normal subgroup | |FULL LIST, MORE INFO | ||
| normal subgroup of finite index | normal subgroup of finite index in the whole group. | |FULL LIST, MORE INFO | ||
| normal subgroup contained in the hypercenter | |FULL LIST, MORE INFO |