Difference between revisions of "Fusion-trivial p-groups"
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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]]]. | 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 | + | 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. |
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| + | 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: | ||
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| + | <div class="boxed"> | ||
| + | === Question on fusion-trivial p-groups === | ||
| + | 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> | ||
Latest revision as of 15:50, 2 May 2024
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 [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 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 [St02] in terms of fusion systems for groups, and for arbitrary saturated fusion systems in [St06].
Theorem 4.8 of [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 [Ma86] (and [HM07]), Henn and Priddy proved in [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 [Th93] it was proved that the [math]p[/math]-groups considered in [HP94] have a strongly characteristic central series, in which each term is the unique subgroup of its isomorphism type. Hence in the sense of [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:
Question on fusion-trivial p-groups
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]?
