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On the point-stabiliser in a transitive permutation groupPotočnik, Primož, 1971- ; Wilson, Steve, matematikZa množico ▫$V$▫, permutacijsko grupo ▫$G$▫ na ▫$V$▫, točko ▫$v \in V$▫ in orbito ▫$\Omega$▫ stabilizatorja ▫$G_v$▫, naj ▫$G_v^\Omega$▫ označuje permutacijsko grupo, ki jo ▫$G_v$▫ inducira na ... ▫$\Omega$▫. Nadalje, naj bo ▫$N$▫ normalizator grupe ▫$G$▫ v simetrični grupi ▫${\rm Sym}(V)$▫. Članek govori o odnosu med grupama ▫$G_v$▫ in ▫$G_v^\Omega$▫. Če je ▫$G$▫ primitivna grupa in ▫$G_v$▫ končna grupa, tedaj sledi iz izreka avtorjev Betten, Delandtsheer, Niemeyer, and Praeger (J. Group Theory 6: 415--420, 2003), da je vsak kompozicijski faktor grupe ▫$G_v$▫ tudi kompozicijski faktor grupe ▫$G_v^{\Omega}$▫. V članku ob nekaterih dodatnih predpostavkah o permutacijski topologiji na ▫${\rm Sym}(V)$▫ ta rezultat posplošimo na neprimitivne grupe ▫$G$▫ z neskončnim stabilizatorjem ▫$G_v$▫. Dokažemo namreč naslednje: Če je ▫$\Omega = u^{G_v}$▫ podorbita tranzitivne zaprte podgrupe ▫$G$▫ v ▫${\rm Sym}(V)$▫ in je ▫$G$▫ kot edinka vsebovana v kaki podgrupi ▫$N$▫ v ▫${\rm Sym}(V)$▫, za katero je ▫$N$▫-orbitala ▫$\{(v^g,u^g) : u\in \Omega, g\in N\}$▫ krepko povezan lokalno končen graf na množici vozlišč ▫$V$▫, tedaj je vsak vsaka zaprta enostavna sekcija grupe ▫$G_v$▫ hkrati tudi sekcija grupe ▫$G_v^{\Omega}$▫. S primerom pokažemo, da topološke predpostavke o grupi ▫$G$▫ in sekcijah ▫$G_v$▫ ne moremo opustiti. Najdemo namreč grupo ▫$G$▫, ki deluje ločno tranzitivno na neskončnem kubičnem drevesu, za katero je stabilizator ▫$G_v$▫ izomorfen modularni grupi ▫$PSL(2,\mathbb{Z}) \cong C_2 \ast C_3$▫, za katero se ve, da ima neskončno mnogo končnih enostavnih grup med svojimi sekcijami.Source: Monatshefte für Mathematik. - ISSN 0026-9255 (Vol. 166, no. 3-4, 2012, str. 947-504)Type of material - article, component partPublish date - 2012Language - englishCOBISS.SI-ID - 15861081
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Potočnik, Primož, 1971- | 18838 |
Wilson, Steve, matematik | ![]() |
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