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KT-fields and sharply triply transitive groups. (English. Russian original) Zbl 1426.20005
Algebra Logic 57, No. 2, 153-160 (2018); translation from Algebra Logika 57, No. 2, 232-242 (2018).
Let \(G\) be a sharply \(3\)-transitive permutation group and let \(C\) be a conjugacy class of involutions of the stabilizer \(G_{\{x,y\}}\) of a \(2\)-set \(\{x,y\}\) of points. The authors prove the following results. If \(cd\) has finite order for all \(c,d\in C\), then the group \(G\) is locally finite (hence known by O. H. Kegel [Arch. Math. 18, 337–348 (1967; Zbl 0189.31103)]). If any two non-commuting elements of \(C\) are conjugate by some element of \(C\), then \(G= \mathrm{PGL}_2 F\) for some field \(F\).
MSC:
20B22 Multiply transitive infinite groups
12K05 Near-fields
20F50 Periodic groups; locally finite groups
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