Abnormal subgroup
This article is about a definition in group theory that is standard among the group theory community (or subcommunity that dabbles in such things) but is not very basic or common for people outside.
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This is an opposite of normality
History
Origin
This term was introduced by: Carter
The notion of abnormal subgroup was introduced by Roger W. Carter in his attempts to understand the structure of Carter subgroups of a solvable group.
Definition
Definition with symbols
A subgroup of a group is termed abnormal if it satisfies the following equivalent conditions:

 (Right action convention): For any in , lies inside the subgroup . Here denotes the conjugate subgroup .
 (Left action convention): For any , we have .
 is a weakly abnormal subgroup of and is not contained in the intersection of two distinct conjugate subgroups.
Relation with other properties
Weaker properties
 Selfnormalizing subgroup
 Weakly abnormal subgroup
 Subabnormal subgroup
 Pronormal subgroup
 Weakly pronormal subgroup
 Paranormal subgroup
 Polynormal subgroup
 Contranormal subgroup
Opposites
The only subgroup of a group that is both normal and abnormal is the whole group itself.
Metaproperties
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition  View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition  View facts about intermediate subgroup condition
If is abnormal inside , is also abnormal inside for any intermediate subgroup .
Upwardclosedness
This subgroup property is upwardclosed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upwardclosed subgroup properties
If is abnormal inside , then so is any subgroup of containing .
Transitivity
NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitiveView variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitiveView facts related to transitivity of subgroup properties  View a survey article on disproving transitivity
The property of being abnormal is not transitive. Its subordination is the property of being subabnormal.
Trimness
The property of being abnormal is identitytrue, that is, any group is abnormal as a subgroup of itself. It is not true for the trivial subgroup unless the whole group is trivial.
Testing
GAP code
One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.GAPcodable subgroup property
View the GAP code for testing this subgroup property at: IsAbnormal
View other GAPcodable subgroup properties  View subgroup properties with inbuilt commands
There's no inbuilt function to test for abnormality, but a short snippet of code can be used to test if a subgroup is abnormal. The function is invoked as follows:
IsAbnormal(group,subgroup);
References
Journal references
 Nilpotent selfnormalizing subgroups of soluble groups by Roger W. Carter, Math. Zeitschr., Volume 75, Page 136  139(Year 1961): ^{Weblink}^{More info}
 Nilpotent selfnormalizing subgroups and system normalizers by Roger W. Carter, Volume 12, Page 535  563(Year 1962): ^{Weblink}^{More info}
 Nilpotent subgroups of finite soluble groups by John S. Rose, Math. Zeitschr., Volume 106, Page 97  112(Year 1968): ^{Weblink}^{More info}
External links
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