# Fusion-trivial p-groups

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 $G=O_{p'}(G)P$). These groups appear in [vdW91].

There does not seem to be any name given to p-groups $P$ for which the only saturated fusion system is $\mathcal{F}_P(P)$. 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 $P$ for which ${\rm Aut}(P)$ is a 2-group, i.e., those abelian 2-groups whose cyclic direct factors have pairwise distinct orders.

A p-group $P$ is fusion-trivial if and only if it is resistant and ${\rm Aut}(P)$ is a p-group. Recall that $P$ is resistant if whenever $\mathcal{F}$ is a saturated fusion system on $P$, we have $\mathcal{F}=N_{\mathcal{F}}(P)$, or equivalently $\mathcal{F}=\mathcal{F}_P(G)$ for some finite group $G$ with $P$ 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 $P$ to be resistant: there exists a central series $P=P_n \geq P_{n-1} \geq \cdots \geq P_1$ for which each $P_i$ is weakly $\mathcal{F}$-closed in any saturated fusion system on $P$. This happens for example if each $P_i$ is the unique subgroup of $P$ of its isomorphism type.

Following [Ma86] (and [HM07]), Henn and Priddy proved in [HP94] that in some sense asymptotically most $p$-groups only occur as Sylow $p$-subgroups of $p$-nilpotent groups. In [Th93] proved that the $p$-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 $p$-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 $p^n$ that are fusion-trivial tend to 1 as $n \rightarrow \infty$?

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