## On groups with a splitting automorphism of prime order.(English. Russian original)Zbl 0841.20030

Sib. Math. J. 34, No. 2, 360-362 (1993); translation from Sib. Mat. Zh. 34, No. 2, 180-183 (1993).
An automorphism $$\varphi$$ of a group $$G$$ is called a splitting automorphism of prime order $$p$$ if $$\varphi^p=1$$, $$x\cdot x^\varphi\cdot x^{\varphi^2}\cdot\dots\cdot x^{\varphi^{p-1}}=1$$ for all $$x\in G$$. For $$\varphi=1$$ this is exactly the definition of a group of prime exponent $$p$$. If $$G$$ is a finite $$p'$$-group then such an automorphism $$\varphi$$ is regular.
Let $${\mathfrak M}_p$$ be the variety of groups with operators, consisting of all groups with a splitting automorphism $$\varphi$$ of prime order $$p$$. The author [Mat. Sb., Nov. Ser. 130(172), No. 1(5), 120-127 (1986; Zbl 0608.20025)] has proved that the nilpotency class of a $$d$$-generator nilpotent group of $${\mathfrak M}_p$$ is bounded by a function depending only on $$d$$ and $$p$$. Thus the locally nilpotent groups in $${\mathfrak M}_p$$ form a subvariety $$LN{\mathfrak M}_p$$. Since the variety $${\mathfrak M}_p$$ is closely connected with classes of groups of prime exponent $$p$$ and groups with regular automorphism of prime order $$p$$, the following question arises: Is it true that the subvariety $$LN{\mathfrak M}_p$$ is a join of the subvariety $${\mathfrak B}\cap LN{\mathfrak M}_p$$ of groups of prime exponent and the subvariety $${\mathfrak N}_{c(p)}\cup{\mathfrak M}_p$$ of nilpotent groups of some $$p$$-bounded class? If the answer to the question is affirmative, then the following equalities must hold in any group $$G$$ in $$LN{\mathfrak M}_p$$: $$(\gamma_{c(p)+1}(G))^p=1$$, $$\gamma_{c(p)+1}(G^p)=1$$. In the present article the author proves that there exist $$p$$-bounded numbers $$k(p)$$ and $$l(p)$$ such that every group $$G$$ in $$LN{\mathfrak M}_p$$ satisfies the identities $$[x^{p^k(p)}_1,x^{p^k(p)}_2,\dots,x^{p^k(p)}_{h(p)+1}]=1$$ which means that the subgroup $$G^{p^{k(p)}}$$ is nilpotent of class $$h(p)$$; i.e., $$\gamma_{h(p)+1}(G^{p^{k(p)}})=1$$ and $$[x_1,x_2,\dots,x_{h(p)+1}]^{p^{l(p)}}=1$$, where $$h(p)$$ is the Higman function bounding the nilpotency class of a nilpotent group with a regular automorphism of prime order $$p$$.

### MSC:

 20E10 Quasivarieties and varieties of groups 20E36 Automorphisms of infinite groups 20D45 Automorphisms of abstract finite groups 20D15 Finite nilpotent groups, $$p$$-groups 20F19 Generalizations of solvable and nilpotent groups 20E25 Local properties of groups

Zbl 0608.20025
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### References:

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