Abnormal subgroup

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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, for any $x$ in $G$, $x$ lies inside the subgroup $<H,H^x>$.

Relation with other properties

Weaker properties

• Self-normalizing subgroup
• Weakly abnormal subgroup
• Subabnormal subgroup
• Pronormal subgroup
• Weakly pronormal subgroup
• Paranormal subgroup
• Polynormal 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.
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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
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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.

References

• Nilpotent self-normalizing subgroups of soluble groups by Roger W. Carter, Math. Zeitschr. 75, 136-139 (1961)
• Nilpotent subgroups of finite soluble groups by John S. Rose, Math. Zeitschr. 106, 97-112 (1968)