# Fusion-trivial p-groups

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] 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]?

In practice, a more realistic question would mimic the asymptotic results mentioned above.