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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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References:
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