Groups with elementary Abelian centralizers of involutions

An involution i of a group G is said to be almost perfect in G if any two involutions of i^G the order of a product of which is infinite are conjugated via a suitable involution in i^G. We generalize a known result by Brauer, Suzuki, and Wall concerning the structure of finite groups with elementary Abelian centralizers of involutions to groups with almost perfect involutions. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 75–82, January–February, 2007.     

Finite groups with 2-closed or 2′-closed centralizers of involutions

Finite nonsolvable groups are described whose involution centralizers are 2-closed or 2-closed, whereas the Sylow p-subgroups for p>2 are cyclic.Translated from Matematicheskie Zametki, Vol. 10, No. 4, pp. 437–446, October, 1971.In conclusion, the author expresses his gratitude to A. I. Starostin for posing the problem and for scientific guidance.     

Supersolvable Frobenius groups with nilpotent centralizers

Let FH be a supersolvable Frobenius group with kernel F and complement H. Suppose that a finite group G admits FH as a group of automorphisms in such a manner that $C_G(F)=1$ and $C_G(H)$ is nilpotent of class c. We show that G is nilpotent of $(c,\left|FH\right|)$-bounded class.     

Finite groups with small centralizers of word-values

Given a positive integer m and a group-word w, we consider a finite group G such that $$w(G) \ne 1$$ and all centralizers of non-trivial w-values have order at most m. We prove that if $$w=v(x_1^{q_1},\dots ,x_k^{q_k})$$, where v is a multilinear commutator word and $$q_1, \dots , q_k$$ are p-powers for some prime p, then the order of G is bounded in terms of w and m only. Similar results hold when w is the nth Engel word or the word $$w=[x^n, y_1, \dots ,y_k]$$ with $$k \ge 1$$.     

Characterization of generalized Chernikov groups among groups with involutions

The class of generalized Chernikov groups is characterized, i.e., the class of periodic locally solvable groups with the primary ascending chain condition. The name of the class is related to the fact that the structure of such groups is close to that of Chernikov groups. Namely, a Chernikov group is defined as a finite extension of a direct product of finitely many quasi-cyclic groups, and a generalized Chernikov group is a layer-finite extension of a direct productA of quasi-cyclicp-groups with finitely many factors for each primep such that each of its elements does not commute elementwise with only finitely many Sylow subgroups ofA. A theorem that characterizes the generalized Chernikov groups in the class of groups with involution is proved. Translated fromMatematicheskie Zametki, Vol. 62, No 4, pp. 577–587, October, 1997. Translated by A. I. Shtern     

Topological groups with invariant centralizers of elements

We consider topological groups in which the centralizer of any element is invariant. We prove that these and only these groups are 2-Engel groups.Translated from Matematicheskie Zametki, Vol. 21, No. 3, pp. 297–300, March, 1977.