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**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 \(p\).

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 \(p\).

Reviewer: N.Yu.Makarenko (Novosibirsk)

### 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 |

### Keywords:

splitting automorphism of prime order; variety of groups; nilpotency class; \(d\)-generator nilpotent groups; locally nilpotent groups; groups of prime exponent; groups with regular automorphism### Citations:

Zbl 0608.20025
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\textit{E. I. Khukhro}, Sib. Math. J. 34, No. 2, 360--362 (1993; Zbl 0841.20030); translation from Sib. Mat. Zh. 34, No. 2, 180--183 (1993)

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### References:

[1] | E. I. Khukhro, ?Locally nilpotent groups admitting a splitting automorphism of prime order,? Mat. Sb.,130, No. 1, 120-127 (1986). · Zbl 0608.20025 |

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[3] | D. R. Hughes and J. G. Thompson, ?TheH p -problem and the structure ofH p -groups,? Pacific J. Math.,9, 1097-1101 (1959). · Zbl 0098.25201 |

[4] | O. H. Kegel, ?Die Nilpotenz derH p -Gruppen,? Math. Z.,75, 373-376 (1960/61). · Zbl 0104.24904 |

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[7] | E. I. Khukhro, ?Finitep-groups admitting an automorphism of orderp with a small number of fixed points,? Mat. Zametki,38, No. 5, 652-657 (1985). |

[8] | G. Higman, ?Groups and rings which have automorphisms without nontrivial fixed elements,? J. London Math. Soc.,32, 321-334 (1957). · Zbl 0079.03203 |

[9] | V. A. Kreknin and A. I. Kostrikin, ?Lie algebras with regular automorphisms,? Dokl. Akad. Nauk SSSR,149, 249-251 (1963). |

[10] | V. A. Kreknin, ?The solubility of Lie algebras with regular automorphisms of finite period,? Dokl. Akad. Nauk SSSR,150, 467-469 (1963). · Zbl 0134.03604 |

[11] | E. I. Khukhro, ?On the Hughes problem for finitep-groups,? Algebra i Logika,26, No. 5, 642-646 (1987). · Zbl 0658.20015 |

[12] | E. I. Khukhro, ?On the structure of finitep-groups admitting a partition,? Sibirsk. Mat. Z.,30, No. 6, 208-218 (1989). |

[13] | E. I. Khukhro, ?Nilpotent groups and their automorphisms of prime order,? to appear in W. de Gruyter (Berlin) series ?Expositions in Mathematics.? |

[14] | N. Yu. Makarenko, ?On almost regular automorphisms of prime order,? Sibirsk. Mat. Z.,33, No. 6, 932-933 (1992). · Zbl 0786.20025 |

[15] | R. B. Warfield, Nilpotent Groups (Lecture Notes in Math.,513), Springer, Berlin (1976). |

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