Journal of Group Theory | 2019
Isomorphism questions for metric ultraproducts of finite quasisimple groups
Abstract
Abstract New results on metric ultraproducts of finite simple groups are established. We show that the isomorphism type of a simple metric ultraproduct of groups X n i \u2062 ( q ) {X_{n_{i}}(q)} ( i ∈ I {i\\in I} ) for X ∈ { PGL , PSp , PGO ( ε ) , PGU } {X\\in\\{\\operatorname{PGL},\\operatorname{PSp},\\operatorname{PGO}^{(\\varepsilon)% },\\operatorname{PGU}\\}} ( ε = ± {\\varepsilon=\\pm} ) along an ultrafilter 𝒰 {\\mathcal{U}} on the index set I for which n i → 𝒰 ∞ {n_{i}\\to_{\\mathcal{U}}\\infty} determines the type X and the field size q up to the possible isomorphism of a metric ultraproduct of groups PSp n i \u2061 ( q ) {\\operatorname{PSp}_{n_{i}}(q)} and a metric ultraproduct of groups PGO n i ( ε ) \u2061 ( q ) {\\operatorname{PGO}_{n_{i}}^{(\\varepsilon)}(q)} . This extends results of [A. Thom and J. Wilson, Metric ultraproducts of finite simple groups, Comp. Rend. Math. 352 2014, 6, 463–466].