On locally nilpotent groups admitting a splitting automorphism of prime order.

*(Russian)*Zbl 0608.20025According to G. Higman [J. Lond. Math. Soc. 32, 321–334 (1957; Zbl 0079.03203)] the nilpotency class of a periodic nilpotent group with a regular automorphism of prime order \(p\) is bounded by some function \(h(p)\). On the other hand, according to A. Kostrikin [Izv. Akad. Nauk SSSR, Ser. Mat. 23, 3–34 (1959; Zbl 0090.24504)] locally nilpotent groups of prime exponent \(p\) form a variety. In other words, for any prime \(p\) and natural \(k\) there exists a number \(c(p,k)\) such that the nilpotency class of any \(k\)-generated nilpotent group of exponent \(p\) doesn’t exceed \(c(p,k)\).

In the present paper these two results are combined in a whole. Namely, it appears that the two classes of groups mentioned above are contained in the variety \({\mathfrak M}_ p\) of groups with operators, and this variety \({\mathfrak M}_ p\) consists of those groups \(G\) that admit a “splitting” automorphism of order \(p\), i.e. an automorphism \(\varphi\) of order \(p\) such that \(x^{\varphi^ 0},x^{\varphi^ 1},...,x^{\varphi^{p-1}}=1\) for all \(x\in G\). The author proves that locally nilpotent groups in \({\mathfrak M}_ p\) form a variety of groups with operators. The deduction of this generalization makes use of the connection between periodic groups and Lie rings, in particular, the techniques from A. Kostrikin’s paper [Izv. Akad. Nauk SSSR, Ser. Mat. 21, 289–310 (1957; Zbl 0080.24601)] are used here. Finite \(p\)-groups from \({\mathfrak M}_ p\) play the main role there, and special efforts are needed to establish the transition from \(p\)-groups to Lie rings and conversely, as the associated Lie ring of a finite \(p\)-group from \({\mathfrak M}_ p\) isn’t necessarily \((p-1)\)-Engel.

In the present paper these two results are combined in a whole. Namely, it appears that the two classes of groups mentioned above are contained in the variety \({\mathfrak M}_ p\) of groups with operators, and this variety \({\mathfrak M}_ p\) consists of those groups \(G\) that admit a “splitting” automorphism of order \(p\), i.e. an automorphism \(\varphi\) of order \(p\) such that \(x^{\varphi^ 0},x^{\varphi^ 1},...,x^{\varphi^{p-1}}=1\) for all \(x\in G\). The author proves that locally nilpotent groups in \({\mathfrak M}_ p\) form a variety of groups with operators. The deduction of this generalization makes use of the connection between periodic groups and Lie rings, in particular, the techniques from A. Kostrikin’s paper [Izv. Akad. Nauk SSSR, Ser. Mat. 21, 289–310 (1957; Zbl 0080.24601)] are used here. Finite \(p\)-groups from \({\mathfrak M}_ p\) play the main role there, and special efforts are needed to establish the transition from \(p\)-groups to Lie rings and conversely, as the associated Lie ring of a finite \(p\)-group from \({\mathfrak M}_ p\) isn’t necessarily \((p-1)\)-Engel.

Reviewer: Uno Kaljulaid (Tartu)

##### MSC:

20F18 | Nilpotent groups |

20E36 | Automorphisms of infinite groups |

20E10 | Quasivarieties and varieties of groups |

20F40 | Associated Lie structures for groups |

20F45 | Engel conditions |

20F50 | Periodic groups; locally finite groups |

20E25 | Local properties of groups |