Lie rings with almost regular automorphisms.
Let \(L\) be a Lie algebra. An automorphism of \(L\) is called regular if it has no fixed point other than 0. It was shown by \textit{V. A. Kreknin} [Sov. Math., Dokl. 4, 683--685 (1963); translation from Dokl. Akad. Nauk SSSR 150, 467--469 (1963; Zbl 0134.03604)] that a Lie ring \(L\) with a regular automorphism of finite order \(k\) is soluble of derived length at most \(2^k-2\). The authors prove the following generalization: Let \(L\) be a Lie algebra with an automorphism \(\varphi\) of order \(p^n\) \((p\) prime) and fixed-point subalgebra \(C_L(\varphi)\) of dimension \(m\). Then \(L\) has a soluble subalgebra \(M\) of derived length bounded by a function of \(p^n\) and codimension bounded by a function of \(m\) and \(p^n\). There is a corresponding result for Lie rings with automorphism where the fixed-point subring is finite, and also an equivalent formulation in terms of graded Lie algebras resp. rings. The proof uses a solubility criterion from [\textit{E. I. Khukhro}, Sib. Math. Zh. 42, 1187--1192 (2001; Zbl 0996.17006)]; it proceeds by a detailed analysis of repeated commutators, as well as the following elementary fact: Let \(V\) be a vector space over a field of prime characteristic \(p\) and \(\psi\) a linear transformation of finite order \(p^r\) whose subspace of fixed points has finite dimension \(m\). Then the dimension of \(V\) is finite and at most \(mp^r\). The paper concludes with an application to groups. A group is said to have finite rank \(r\) if every finitely generated subgroup can be generated by \(r\) elements, and \(r\) is the least such number. If \(G\) is a torsion-free locally nilpotent group, one can form the Malcev completion \(\sqrt G\) consisting of all roots of elements of \(G\) [see e.g. Chapter 10 in \textit{E. I. Khukhro}, \(p\)-automorphisms of finite \(p\)-groups, Cambridge University Press (1998; Zbl 0897.20018)], and the authors first show that if a locally nilpotent torsion-free group \(C\) has finite (Abelian) rank \(r\), then the Lie \(Q\)-algebra \(U\) corresponding to the Malcev completion \(\sqrt C\) has finite dimension bounded in terms of \(r\). (For a related result, see \textit{A. I. Malcev}, Mat. Sb., N. Ser 28(70), 567--588 (1951; Zbl 0043.02301); Am. Math. Soc., Transl. (2) 2, 1--21 (1956)]. They now derive the following Theorem: Let \(G\) be a locally nilpotent torsion-free group with an automorphism \(\varphi\) of order \(p^n\) \((p\) prime) such that the fixed-point subgroup \(C(\varphi)\) has finite rank \(r\). Then \(G\) has a soluble subgroup \(H\) of derived length bounded by a function of \(p^n\) and of finite corank \(t=t(r,p^n)\) in \(G\) bounded above in terms of \(r\) and \(p^n\) in the sense that there are \(t\) elements \(g_1,\dots,g_t\) such that every element of \(G\) is a root of an element of the subgroup \(\langle g_1,\dots,g_t,H\rangle\).
scientific article; zbMATH DE number 1956755
scientific article