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In this paper we study sharply 3-transitive groups. The local finiteness of sharply triply transitive permutation groups of characteristic p > 3 containing a finite element of order p is proved. Keywords: group, sharply k-transitive group, sharply 3-transitive group, locally finite group, neardomain, near-field

We study groups with an H-Frobenius element and the nilpotent kernels of the corresponding Frobenius subgroups. We prove the two theorems that solve Question 10.61 of The Kourovka Notebook in this case.

We study the structure of an infinite group with automorphism of order 2p, where p is an odd prime leaving only the identity element fixed.

We prove Jordan’s Theorem for infinite sharply 2-transitive groups satisfying the finiteness (a, b)-condition, with |a| · |b| even.

We prove local finiteness of sharply triply transitive permutation groups with periodic two-point stabilizers.

We prove that an isolated involution of an infinite group does not always belong to the preimage of the center of the quotient group by the maximal (periodic) subgroups without involution.

Glauberman’s Z *-theorem [1] and the theorem of Bender are two most important tools for local analysis in the theory of finite groups. The Z *-theorem generalizes the known Burnside and Brauer–Suzuki theorems on finite groups with cyclic and quaternion Sylow 2-subgroups. Whether these theorems are valid in a class of periodic groups is unknown. We prove that the Z *-theorem is invalid in the class of all periodic groups. In particular, this gives negative answers to questions of A. V. Borovik [3, Question 11.13] and V. D. Mazurov [3, Question 17.71a].

This work contains a survey of some basic results on finite projective planes which were obtained by a group of mathematicians from Krasnoyarsk since 1985.