Abnormal subgroup

This article is about a definition in group theory that is standard among the group theory community (or sub-community 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]
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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 $H$ of a group $G$ is termed abnormal if it satisfies the following equivalent conditions:

- (Right action convention): For any $x$ in $G$ , $x$ lies inside the subgroup $\langle H,H^{x}\rangle$ . Here $H^{x}$ denotes the conjugate subgroup $x^{-1}Hx$ .
- (Left action convention): For any $x\in G$ , we have $x\in \langle H,xHx^{-1}\rangle$ .
$H$ is a weakly abnormal subgroup of $G$ and is not contained in the intersection of two distinct conjugate subgroups.

Relation with other properties

Weaker properties

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.
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If $H$ is abnormal inside $G$ , $H$ is also abnormal inside $K$ for any intermediate subgroup $K$ .

Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upward-closed subgroup properties

If $H$ is abnormal inside $G$ , then so is any subgroup $K$ of $G$ containing $H$ .

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
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The property of being abnormal is not transitive. Its subordination is the property of being subabnormal.

Trimness

The property of being abnormal is identity-true, 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.
View the GAP code for testing this subgroup property at: IsAbnormal
View other GAP-codable subgroup properties | View subgroup properties with in-built commands

GAP-codable subgroup property

There's no in-built 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 self-normalizing subgroups of soluble groups by Roger W. Carter, Math. Zeitschr., Volume 75, Page 136 - 139(Year 1961): ^Weblink^{More info}
Nilpotent self-normalizing 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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