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Contents 1 Definition 1.1 Definition with symbols 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Metaproperties 3.1 Intermediate subgroup condition 3.2 Trimness 3.3 Join-closedness 4 Testing 4.1 GAP code 5 References

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
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a list of other standard non-basic definitions

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | | |
RANDOM SUBGROUP PROPERTY: Central factor: A subgroup with the property that every inner automorphism of the whole group restricts to an inner automorphism of the subgroup.

This is a variation of normality
View a complete list of variations of normality OR read a survey article on varying normality

Definition

Definition with symbols

A subgroup H of a group G is termed paranormal if for any , H is a contranormal subgroup of ; in other words, the normal closure of H in is .

Here H^g = g ^{− 1}Hg is a conjugate of H, and the angled braces are for the subgroup generated.

Relation with other properties

Stronger properties

• Normal subgroup
• Pronormal subgroup: For proof of the implication, refer Pronormal implies paranormal and for proof of its strictness (i.e. the reverse implication being false) refer Paranormal not implies pronormal.
• Abnormal subgroup
• Join of pronormal subgroups
• Strongly paranormal subgroup: For proof of the implication, refer Strongly paranormal implies paranormal and for proof of its strictness (i.e. the reverse implication being false) refer Paranormal not implies strongly paranormal.
• Weakly abnormal subgroup: For proof of the implication, refer Weakly abnormal implies paranormal and for proof of its strictness (i.e. the reverse implication being false) refer Paranormal not implies weakly abnormal.

Weaker properties

• Polynormal subgroup: It has been conjectured that for finite groups, the two notions coincide; however this has neither been proved nor disproved.
• Weakly normal subgroup: For proof of the implication, refer Paranormal implies weakly normal and for proof of its strictness (i.e. the reverse implication being false) refer Weakly normal not implies paranormal.

Metaproperties

Intermediate subgroup condition

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
View all subgroup properties satisfying the intermediate subgroup condition|View facts related to the intermediate subgroup condition

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself)
View all trim subgroup properties OR view trivially true subgroup properties OR view identity-true subgroup properties

Join-closedness

This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property
View a complete list of join-closed subgroup properties

In fact, an arbitrary, possibly empty, join of paranormal subgroups is paranormal. For full proof, refer: Paranormality is strongly join-closed

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: IsParanormal
View other GAP-codable subgroup properties | View subgroup properties with in-built commands

There is no built-in GAP command to test paranormality, but a short piece of GAP code can achieve this. The code is available at GAP:IsParanormal.

References

• On the arrangement of intermediate subgroups by M. S. Ba and Z. I. Borevich
• On the arrangement of subgroups by Z. I. Borevich, Zap. Nauchn. Semin. tOMI, 94, 5-12 (1979)
• On the lattice of subgroups by Z. I. Borevich and O. N. Macedonska, Zap. Nauchn. Semin. LOMI, 103, 13-19, 1980
• Testing of subgroups of a finite group for some embedding properties like pronormality by V. I. Mysovskikh