Finite normal subgroup
This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): finite group
View a complete list of such conjunctions
Definition
A subgroup of a group is termed a finite normal subgroup if it is finite (as a group) and normal as a subgroup.
Examples
VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups dissatisfying property finite group
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| normal subgroup of prime order | normal subgroup and its order is a prime number | |||
| finite central subgroup | finite and a central subgroup | central implies normal | any of the finite examples for normal not implies central | |FULL LIST, MORE INFO |
| finite characteristic subgroup | finite and a characteristic subgroup | any of the finite examples for normal not implies characteristic | |FULL LIST, MORE INFO | |
| normal subgroup of finite group | normal and the whole group is finite | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finitely generated normal subgroup | normal subgroup and finitely generated as a group | |FULL LIST, MORE INFO | ||
| normal closure of finite subset | it is the normal subgroup generated by a finite subset of the whole group | |FULL LIST, MORE INFO | ||
| finite subnormal subgroup | finite and a subnormal subgroup | |FULL LIST, MORE INFO | ||
| periodic normal subgroup | normal subgroup in which every element has finite order | |FULL LIST, MORE INFO | ||
| join-transitively finite subgroup | its join with any finite subgroup of the whole group is finite | any non-normal subgroup of a finite group works | |FULL LIST, MORE INFO | |
| amalgam-normal-subhomomorph-containing subgroup | finite normal implies amalgam-normal-subhomomorph-containing | |FULL LIST, MORE INFO | ||
| amalgam-characteristic subgroup | it is characteristic inside the join of the whole group with itself over the subgroup | finite normal implies amalgam-characteristic (also via amalgam-normal-subhomomorph-containing) | any central subgroup that is not finite works | |FULL LIST, MORE INFO |
| quotient-powering-invariant subgroup | if the group is powered over a prime, so is the quotient group | finite normal implies quotient-powering-invariant | any infinite group as a subgroup of itself works | |FULL LIST, MORE INFO |
| powering-invariant subgroup | if the group is powered over a prime, so is the subgroup | (via quotient-powering-invariant) | (via quotient-powering-invariant) | |FULL LIST, MORE INFO |
| local powering-invariant subgroup | if an element of the subgroup has a unique root in the group, that root is in the subgroup | (via finite) | |FULL LIST, MORE INFO |
