Fusion-trivial p-groups - Revision history

Charles Eaton at 15:50, 2 May 2024

2024-05-02T15:50:10Z

Charles Eaton at 15:48, 2 May 2024

2024-05-02T15:48:23Z

Charles Eaton: [HP94] reference

2022-08-05T13:42:15Z

[HP94] reference

Charles Eaton: Question on asymptotics added

2021-05-25T16:58:41Z

Question on asymptotics added

Charles Eaton: Created page with "A p-group $P$ is ''p-nilpotent forcing'' if any finite group $G$ that contains $P$ as a Sylow p-subgroup must be p-nilpotent (that is $G=..."$

2020-11-18T21:57:55Z

Created page with "A p-group <math>P</math> is ''p-nilpotent forcing'' if any finite group <math>G</math> that contains <math>P</math> as a Sylow p-subgroup must be p-nilpotent (that is <math>G=..."

New page

A p-group <math>P</math> is ''p-nilpotent forcing'' if any finite group <math>G</math> that contains <math>P</math> as a Sylow p-subgroup must be p-nilpotent (that is <math>G=O_{p'}(G)P</math>). These groups appear in [[References#W|[vdW91]]].

There does not seem to be any name given to p-groups <math>P</math> for which the only saturated fusion system is <math>\mathcal{F}_P(P)</math>. We will refer to them as ''fusion-trivial p-groups'' (although appropriate names might also be ''nilpotent forcing'' or ''fusion nilpotent forcing''). Examples of such p-groups are abelian 2-groups <math>P</math> for which <math>{\rm Aut}(P)</math> is a 2-group, i.e., those abelian 2-groups whose cyclic direct factors have pairwise distinct orders.

A p-group <math>P</math> is fusion-trivial if and only if it is [[Glossary|resistant]] and <math>{\rm Aut}(P)</math> is a p-group. Recall that <math>P</math> is resistant if whenever <math>\mathcal{F}</math> is a saturated fusion system on <math>P</math>, we have <math>\mathcal{F}=N_{\mathcal{F}}(P)</math>, or equivalently <math>\mathcal{F}=\mathcal{F}_P(G)</math> for some finite group <math>G</math> with <math>P</math> as a normal Sylow p-subgroup. Resistant p-groups were introduced in [[References#S|[St02]]] in terms of fusion systems for groups, and for arbitrary saturated fusion systems in [[References#S|[St06]]].

Theorem 4.8 of [[References#S|[St06]]] yields a useful necessary and sufficient condition for a p-group <math>P</math> to be resistant: there exists a central series \[P=P_n \geq P_{n-1} \geq \cdots \geq P_1\] for which each <math>P_i</math> is weakly <math>\mathcal{F}</math>-closed in any saturated fusion system on <math>P</math>. This happens for example if each <math>P_i</math> is the unique subgroup of <math>P</math> of its isomorphism type.

← Older revision		Revision as of 15:50, 2 May 2024
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	Does the proportion of p-groups of order <math>p^n</math> that are fusion-trivial tend to 1 as <math>n \rightarrow \infty</math>?		Does the proportion of p-groups of order <math>p^n</math> that are fusion-trivial tend to 1 as <math>n \rightarrow \infty</math>?
	</div>		</div>
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−	~~In practice, a more realistic question would mimic the asymptotic results mentioned above.~~

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 Theorem 4.8 of [[References#S|[St06]]] yields a useful necessary and sufficient condition for a p-group <math>P</math> to be resistant: there exists a central series \[P=P_n \geq P_{n-1} \geq \cdots \geq P_1\] for which each <math>P_i</math> is weakly <math>\mathcal{F}</math>-closed in any saturated fusion system on <math>P</math>. This happens for example if each <math>P_i</math> is the unique subgroup of <math>P</math> of its isomorphism type.
-Following [[References#M|[Ma86]]] (and [[References#H|[HM07]]]), Henn and Priddy proved in [[References#H|[HP94]]] that in some sense asymptotically most <math>p</math>-groups only occur as Sylow <math>p</math>-subgroups of <math>p</math>-nilpotent groups. In [[References#T|[Th93]]] proved that the <math>p</math>-groups considered in [[References#H|[HP94]]] have a strongly characteristic central series, in which each term is the unique subgroup of its isomorphism type. Hence in the sense of [[References#M|[Ma86]]], almost every <math>p</math>-group is fusion trivial. This leaves the natural question of whether a version of this result with a cleaner definition of "almost all" holds:
+Following [[References#M|[Ma86]]] (and [[References#H|[HM07]]]), Henn and Priddy proved in [[References#H|[HP94]]] that in some sense asymptotically most <math>p</math>-groups only occur as Sylow <math>p</math>-subgroups of <math>p</math>-nilpotent groups. In [[References#T|[Th93]]] it was proved that the <math>p</math>-groups considered in [[References#H|[HP94]]] have a strongly characteristic central series, in which each term is the unique subgroup of its isomorphism type. Hence in the sense of [[References#M|[Ma86]]], almost every <math>p</math>-group is fusion trivial. This leaves the natural question of whether a version of this result with a cleaner definition of "almost all" holds:
 <div class="boxed">

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 Theorem 4.8 of [[References#S|[St06]]] yields a useful necessary and sufficient condition for a p-group <math>P</math> to be resistant: there exists a central series \[P=P_n \geq P_{n-1} \geq \cdots \geq P_1\] for which each <math>P_i</math> is weakly <math>\mathcal{F}</math>-closed in any saturated fusion system on <math>P</math>. This happens for example if each <math>P_i</math> is the unique subgroup of <math>P</math> of its isomorphism type.
-With reference to the result to which [[References#G|[GU07]]] aspires in mind, that "the automorphism group of a finite p-group is almost always a p-group", it's natural to ask the following question:
+Following [[References#M|[Ma86]]] (and [[References#H|[HM07]]]), Henn and Priddy proved in [[References#H|[HP94]]] that in some sense asymptotically most <math>p</math>-groups only occur as Sylow <math>p</math>-subgroups of <math>p</math>-nilpotent groups. In [[References#T|[Th93]]] proved that the <math>p</math>-groups considered in [[References#H|[HP94]]] have a strongly characteristic central series, in which each term is the unique subgroup of its isomorphism type. Hence in the sense of [[References#M|[Ma86]]], almost every <math>p</math>-group is fusion trivial. This leaves the natural question of whether a version of this result with a cleaner definition of "almost all" holds:
 <div class="boxed">
 === Question on fusion-trivial p-groups ===
-Is almost every finite p-group fusion trivial? I.e., does the proportion of p-groups of order <math>p^n</math> that are fusion-trivial tend to 1 as <math>n \rightarrow \infty</math>?
+Does the proportion of p-groups of order <math>p^n</math> that are fusion-trivial tend to 1 as <math>n \rightarrow \infty</math>?
 </div>
-In practice, a more realistic question would mimic the asymptotic results of [[References#G|[GU07]]].
+In practice, a more realistic question would mimic the asymptotic results mentioned above.