Normal subgroup of finite group
From Groupprops
This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property imposed on the ambient group: finite group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
Contents
Definition
A subgroup of a group is termed a normal subgroup of finite group if it satisfies the following equivalent conditions:
- The whole group is a finite group and the subgroup is a normal subgroup of it.
- There exists a finite group containing such that is a characteristic subgroup of .
- is a finite group and there is a generating set for and a generating set for such that the conjugate of any element of by any element of is in .
Equivalence of definitions
- The equivalence of definitions (1) and (2) follows from the finite NPC theorem.
- PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Examples
VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups whose group part dissatisfies property finite group
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions
Relation with other properties
How it differs from normality in general
The property of normality inside a finite group is largely similar to the property of normality in an arbitrary group -- most of the property implications, metaproperties, etc. hold for finite groups. There are some small differences:
- To check that a subgroup of a finite group is normal, it suffices to check that it is sent to within itself under all elements in a generating set of the whole group. For an infinite group, we need to check that it is sent to within itself by all elements of a generating set and their inverses.
- Normal subgroups of finite groups, and more generally finite normal subgroups, behave nicely with respect to amalgams: finite normal implies amalgam-characteristic. This is not true for normal subgroups in arbitrary groups.
Stronger properties
property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|
Characteristic subgroup of finite group | characteristic implies normal | normal not implies characteristic | |FULL LIST, MORE INFO | |
Fully invariant subgroup of finite group | ||||
Isomorph-free subgroup of finite group | ||||
Subgroup of finite abelian group | ||||
Normal subgroup of group of prime power order |
Weaker properties
property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|
Finite normal subgroup | |FULL LIST, MORE INFO | |||
Finitely generated normal subgroup | ||||
Normal closure of finite subset | ||||
Normal subgroup of finite index | ||||
Normal subgroup of periodic group | ||||
Normal subgroup of finitely generated group | ||||
Normal subgroup of slender group |