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On locally nilpotent groups admitting a splitting automorphism of prime order. (Russian) Zbl 0608.20025
According 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.

##### 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
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