Fixed points of automorphisms of Lie rings and locally finite groups

Finite groups and the fixed points of coprime automorphisms

“…Later the author found that if, under these assumptions, C G (a) is nilpotent of class at most c (respectively C G (a) is nilpotent of class at most c) for any a ∈ A # , and if G has derived length d, then the nilpotency class of G (respectively of G ) is {c, d, p}-bounded [6]. In the present paper we show that actually much stronger results are valid: the bounds on the class of G and G can be chosen independent of d. A well-known theorem of Kreknin [5] says that if a Lie ring L admits a fixedpoint-free automorphism of finite order n, then L is solvable and the derived length of L is bounded by a function of n. We will require the following extension of this result [4]. …”

Almost regular automorphisms

“…He raised the question whether there is a function of m and n bounding the m derived length of such a Lie algebra. An effective proof (valid also for infinite-dimensional algebras) was found by Khukhro and Shumyatsky in [53]: induction on s = 1, 2,..., n -1 proves simultaneously that …”

Automorphisms of finite p-groups admitting a partition

“…We state for convenience of the reader two main results of [5]. Recall that a quantity is said to be (a, b, .…”

“…In this note we extend to finite p-groups with a partition the following result on groups of prime exponent p due to the author and Shumyatsky [5]: if a finite group G of exponent p admits a soluble group of automorphisms A of coprime order such that the fixed-point subgroup C G (A) is soluble of derived length d, then G is nilpotent of (p, d, |A|)-bounded class. Examples show that the nilpotency class of the whole group P cannot be bounded.…”

“…Examples show that the nilpotency class of the whole group P cannot be bounded. The bound for the nilpotency class of that maximal subgroup can be chosen to be the same as the bound in the aforementioned result of [5] for groups of exponent p. The proof is based on the theorem in [5] for groups of exponent p and the method of "elimination of automorphisms by nilpotency", which was developed earlier by the author, in particular, for studying finite p-groups with a splitting automorphism of order p; see [6] and [4,Ch. 6].…”