View source for Paranormal subgroup

{{stdnonbasicdef}}
{{subgroup property}}
{{variation of|normality}}

==Definition==

===Definition with symbols===

(''right-action convention''): A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''paranormal''' if for any <math>g \in G</math>, <math>H</math> is a [[defining ingredient::contranormal subgroup]] of <math>\langle H, H^g \rangle</math>; in other words, the normal closure of <math>H</math> in <math>\langle H, H^g \rangle</math> is <math>\langle H, H^g \rangle</math>.

Here <math>H^g = g^{-1}Hg</math> is a [[conjugate subgroups|conjugate]] of <math>H</math>, and the angled braces are for the subgroup generated.

(''left-action convention''): A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''paranormal''' if for any <math>g \in G</math>, <math>H</math> is a [[defining ingredient::contranormal subgroup]] of <math>\langle H, gHg^{-1} \rangle</math>; in other words, the normal closure of <math>H</math> in <math>\langle H, gHg^{-1} \rangle</math> is <math>\langle H, gHg^{-1} \rangle</math>.

==Relation with other properties==

===Stronger properties===

* [[Weaker than::Normal subgroup]]
* [[Weaker than::Pronormal subgroup]]: {{proofofstrictimplicationat|[[Pronormal implies paranormal]]|[[Paranormal not implies pronormal]]}}
* [[Weaker than::Abnormal subgroup]]
* [[Weaker than::Join of pronormal subgroups]]
* [[Weaker than::Strongly paranormal subgroup]]: {{proofofstrictimplicationat|[[Strongly paranormal implies paranormal]]|[[Paranormal not implies strongly paranormal]]}}
* [[Weaker than::Weakly abnormal subgroup]]: {{proofofstrictimplicationat|[[Weakly abnormal implies paranormal]]|[[Paranormal not implies weakly abnormal]]}}
* [[Weaker than::Paracharacteristic subgroup]]
* [[Weaker than::Sylow subgroup]]
* [[Weaker than::Sylow subgroup of normal subgroup]]
* [[Weaker than::Hall subgroup]]: {{proofat|[[Hall implies paranormal]]}}
* [[Weaker than::Hall subgroup of normal subgroup]]: {{proofat|[[Hall of normal implies paranormal]]}}
* [[Weaker than::Intermediately isomorph-conjugate subgroup]]
* [[Weaker than::Procharacteristic subgroup]]

===Weaker properties===

* [[Stronger than::Polynormal subgroup]]: It has been conjectured that for [[finite group]]s, the two notions coincide; however this has neither been proved nor disproved.
* [[Stronger than::Weakly normal subgroup]]: {{proofofstrictimplicationat|[[Paranormal implies weakly normal]]|[[Weakly normal not implies paranormal]]}}
* [[Stronger than::Intermediately subnormal-to-normal subgroup]]
* [[Stronger than::Subnormal-to-normal subgroup]]

==Metaproperties==

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{{join-closed}}

In fact, an arbitrary, possibly empty, join of paranormal subgroups is paranormal. {{proofat|[[Paranormality is strongly join-closed]]}}

==Testing==

{{GAP code for subgroup property|test = IsParanormal}}

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