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E. I. Khukhro

Publications and source records attributed to E. I. Khukhro.

27 records · Page 2Linked to original sources
arXiv2015

On finite soluble groups with almost fixed-point-free automorphisms of non-coprime order

It is proved that if a finite $p$-soluble group $G$ admits an automorphism $φ$ of order $p^n$ having at most $m$ fixed points on every $φ$-invariant elementary abelian $p'$-section of $G$, then the $p$-length of $G$ is bounded above in terms of $p^n$ and $m$; if in addition the group $G$ is soluble, then the Fitting height of $G$ is bounded above in terms of $p^n$ and $m$. It is also proved that if a finite soluble group $G$ admits an automorphism $ψ$ of order $p^aq^b$ for some primes $p,q$, then the Fitting height of $G$ is bounded above in terms of $|ψ|$ and $|C_G(ψ)|$.

math.GR↗
arXiv2014

Finite p-groups with a Frobenius group of automorphisms whose kernel is a cyclic p-group

Suppose that a finite $p$-group $P$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ that is a cyclic $p$-group and with complement $H$. It is proved that if the fixed-point subgroup $C_P(H)$ of the complement is nilpotent of class $c$, then $P$ has a characteristic subgroup of index bounded in terms of $c$, $|C_P(F)|$, and $|F|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. Examples show that the condition of $F$ being cyclic is essential. The proof is based on a Lie ring method and a theorem of the authors and P. Shumyatsky about Lie rings with a metacyclic Frobenius group of automorphisms $FH$. It is also proved that $P$ has a characteristic subgroup of $(|C_P(F)|, |F|)$-bounded index whose order and rank are bounded in terms of $|H|$ and the order and rank of $C_P(H)$, respectively, and whose exponent is bounded in terms of the exponent of $C_P(H)$.

math.GR↗
arXiv2014

Words and pronilpotent subgroups in profinite groups

Let $w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.

math.GR↗
arXiv2014

Nonsoluble and non-p-soluble length of finite groups

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $λ(G)$ as the minimum number of nonsoluble factors in a series of this kind. Upper bounds for $λ(G)$ appear in the study of various problems on finite, residually finite, and profinite groups. We prove that $λ(G)$ is bounded in terms of the maximum $2$-length of soluble subgroups of $G$, and that $λ(G)$ is bounded by the maximum Fitting height of soluble subgroups. For an odd prime $p$, the non-$p$-soluble length $λ_p(G)$ is introduced, and it is proved that $λ_p(G)$ does not exceed the maximum $p$-length of $p$-soluble subgroups. We conjecture that for a given prime $p$ and a given proper group variety ${\frak V}$ the non-$p$-soluble length $λ_p(G)$ of finite groups $G$ whose Sylow $p$-subgroups belong to ${\frak V}$ is bounded. In this paper we prove this conjecture for any variety that is a product of several soluble varieties and varieties of finite exponent.

math.GR↗
arXiv2014

On the length of finite factorized groups

The nonsoluble length $λ(G)$ of a finite group $G$ is defined as the number of nonsoluble factors in a shortest normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where $F^*_1(G)=F^*(G)$ is the generalized Fitting subgroup, and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. It is proved that if a finite group $G=AB$ is factorized by two subgroups of coprime orders, then the nonsoluble length of $G$ is bounded in terms of the generalized Fitting heights of $A$ and $B$. It is also proved that if, say, $B$ is soluble of derived length $d$, then the generalized Fitting height of $G$ is bounded in terms of $d$ and the generalized Fitting height of $A$.

math.GR↗
arXiv2014

Nereshennye voprosy teorii grupp. Kourovskaya tetrad'. 18-e izd. (Russian)

This is a collection of open problems in Group Theory proposed by more than 300 mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965, now also in English. This is the 18th Russian edition, which contains 120 new problems and a number of comments on about 1000 problems from the previous editions.

math.GR↗
arXiv2013

Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms

Suppose that a finite group $G$ admits a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H such that the fixed point subgroup $C_G(H)$ of the complement is nilpotent of class $c$. It is proved that $G$ has a nilpotent characteristic subgroup of index bounded in terms of $c$, $|C_G(F)|$, and $|FH|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. This generalizes the previous theorem of the authors and P. Shumyatsky, where for the case of $C_G(F)=1$ the whole group was proved to be nilpotent of $(c,|H|)$-bounded class. Examples show that the condition of $F$ being cyclic is essential. B. Hartley's theorem based on the classification provides reduction to soluble groups. Then representation theory arguments are used to bound the index of the Fitting subgroup. Lie ring methods are used for nilpotent groups. A similar theorem on Lie rings with a metacyclic Frobenius group of automorphisms $FH$ is also proved.

math.GR↗
arXiv2013

Lie algebras admitting a metacyclic Frobenius group of automorphisms

Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ such that the characteristic of the ground field does not divide $|H|$. It is proved that if the subalgebra $C_L(F)$ of fixed points of the kernel has finite dimension $m$ and the subalgebra $C_L(H)$ of fixed points of the complement is nilpotent of class $c$, then $L$ has a nilpotent subalgebra of finite codimension bounded in terms of $m$, $c$, $|H|$, and $|F|$ whose nilpotency class is bounded in terms of only $|H|$ and $c$. Examples show that the condition of the kernel $F$ being cyclic is essential.

math.RA↗