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 isproved that if the fixed-point subgroup $C_P(H)$ of the complement is nilpotentof class $c$, then $P$ has a characteristic subgroup of index bounded in termsof $c$, $|C_P(F)|$, and $|F|$ whose nilpotency class is bounded in terms of $c$and $|H|$ only. Examples show that the condition of $F$ being cyclic isessential. The proof is based on a Lie ring method and a theorem of the authorsand P. Shumyatsky about Lie rings with a metacyclic Frobenius group ofautomorphisms $FH$. 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 isbounded in terms of the exponent of $C_P(H)$.
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机译：假设一个有限的$ p $ -group $ P $接受一个Frobenius自同构$ FH $组，其内核$ F $是一个循环$ p $ -group，并且补语$ H $。证明如果补数的定点子组$ C_P（H）$为类$ c $的零，则$ P $具有索引的特征子组，其索引范围为$ c $，$ | C_P（F）| $以及$ | F | $的幂等性类仅以$ c $和$ | H | $为边界。实例表明，$ F $是循环必要条件。证明基于李环方法和作者以及P. Shumyatsky关于带有自环同构亚基的Frobenius群的李环的一个定理。还证明了$ P $具有$（| C_P（F）|，| F |）$有界索引的特征子组，其顺序和等级受$ | H | $以及$ C_P（H）$，其指数以$ C_P（H）$的指数为界。
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