具有有界中心化子链的群与~ Borovik-Khukhro猜想
Groups with bounded centralizer chains and the~Borovik--Khukhro conjecture
摘要：
设$G$是局部有限群，$F(G)$是$G$的Hirsch-Plotkin根.用$S$表示$G/F(G)$的广义Fitting子群在$G$中的全逆像。假设存在一个数字$k$，使得$G$中的嵌套中心器的每个链的长度不超过$k$。Borovik-Khukhro猜想指出，特别是，在这个假设下的商$G/S$包含一个交换子群的指数有界的$k$。我们反驳了这个说法，并证明了它的一些较弱的模拟。
英文摘要：
Let $G$ be a locally finite group and $F(G)$ the Hirsch--Plotkin radical of $G$. Denote by $S$ the full inverse image of the generalized Fitting subgroup of $G/F(G)$ in $G$. Assume that there is a number $k$ such that the length of every chain of nested centralizers in $G$ does not exceed $k$. The Borovik--Khukhro conjecture states, in particular, that under this assumption the quotient $G/S$ contains an abelian subgroup of index bounded in terms of $k$. We disprove this statement and prove some its weaker analog.
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