Difference between revisions of "Fusiontrivial pgroups"
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Revision as of 22:57, 18 November 2020
A pgroup [math]P[/math] is pnilpotent forcing if any finite group [math]G[/math] that contains [math]P[/math] as a Sylow psubgroup must be pnilpotent (that is [math]G=O_{p'}(G)P[/math]). These groups appear in [vdW91].
There does not seem to be any name given to pgroups [math]P[/math] for which the only saturated fusion system is [math]\mathcal{F}_P(P)[/math]. We will refer to them as fusiontrivial pgroups (although appropriate names might also be nilpotent forcing or fusion nilpotent forcing). Examples of such pgroups are abelian 2groups [math]P[/math] for which [math]{\rm Aut}(P)[/math] is a 2group, i.e., those abelian 2groups whose cyclic direct factors have pairwise distinct orders.
A pgroup [math]P[/math] is fusiontrivial if and only if it is resistant and [math]{\rm Aut}(P)[/math] is a pgroup. 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 psubgroup. Resistant pgroups 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 pgroup [math]P[/math] to be resistant: there exists a central series \[P=P_n \geq P_{n1} \geq \cdots \geq P_1\] 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.