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 \(p\).


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
Full Text: DOI


[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
[2] E. I. Khukhro, ?Nilpotency of soluble groups admitting a splitting automorphism of prime order,? Algebra i Logika,19, No. 1, 118-129 (1980). · Zbl 0475.20018
[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
[5] A. I. Kostrikin, ?On a Burnside problem,? Izv. Akad. Nauk SSSR Ser. Mat.,23, No. 1, 3-34 (1959).
[6] J. Alperin, ?Automorphisms of solvable groups,? Proc. Amer. Math. Soc.,13, 175-180 (1962). · Zbl 0104.02801
[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).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.