Almost All Generalized Extraspecial p-Groups Are Resistant

Abstract. In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible Fradical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p-local finite group, getting some new examples of these for p = 3.

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“…• Theorems 4.1, 4.2, 4.3, 5.1 and 5.8 identify a large family of resistant groups complementing the results in [38], and extending those in [32].…”

Section: Theorem 11 Let P Be An Odd Prime and Let S Be A Rank Two Pmentioning

confidence: 60%

All 𝑝-local finite groups of rank two for odd prime 𝑝

Díaz

Ruiz

Viruel³

2006

Trans. Amer. Math. Soc.

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“…So far several families of Swan groups have been found, as for example abelian p-groups [31], most of the generalized extraspecial p-groups [20,30] or metacyclic p-groups for p odd [12]. In fact, almost all p-groups are Swan groups [25].…”

Section: Control Of Fusionmentioning

confidence: 99%

Cellularization of classifying spaces and fusion properties of finite groups

Flores

Scherer

2007

Journal of the London Mathematical Society

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Abstract. One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their BZ/p-cellularization. In the easiest cases this is a classifying space of a finite group (always a finite p-group). If not, weshow that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears.

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“…If Q is F-essential then Q is F-centric radical. Alperin's fusion theorem implies that F is generated by the automorphism groups Aut F ðQÞ of F-essential subgroups Q of P. Following Stancu [23], the axioms (I-S) and (II-S) imply the a priori stronger axioms used in work of Broto Levi and Oliver [2], where fusion systems are called saturated fusion systems. Remark 2.2.…”

Section: Generalities On Fusion Systemsmentioning

confidence: 99%