Let f(x) be a non-zero polynomial with integer coefficients. An automorphism φ of a group G is said to satisfy the elementary abelian identity f(x) if the linear transformation induced by φ on every ...characteristic elementary abelian section of G is annihilated by f(x). We prove that if a finite (soluble) group G admits a fixed-point-free automorphism φ satisfying an elementary abelian identity f(x), where f(x) is a primitive polynomial, then the Fitting height of G is bounded in terms of deg(f(x)). We also prove that if f(x) is any non-zero polynomial and G is a σ′-group for a finite set of primes σ=σ(f(x)) depending only on f(x), then the Fitting height of G is bounded in terms of the number irr(f(x)) of different irreducible factors in the decomposition of f(x). These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length α(|φ|) of 〈φ〉 when degf(x) or irr(f(x)) is small in comparison with α(|φ|).
Let G be a finite soluble group, and let h(G) be the Fitting length of G. If φ is a fixed-point-free automorphism of G, that is CG(φ)={1}, we denote by W(φ) the composition length of 〈φ〉. A ...long-standing conjecture is that h(G)≤W(φ), and it is known that this bound is always true if the order of G is coprime to the order of φ. In this paper we find some bounds to h(G) in function of W(φ) without assuming that (|G|,|φ|)=1. In particular we prove the validity of the “universal” bound h(G)<7W(φ)2. This improves the exponential bound known earlier from a special case of a theorem of Dade.
We study periodic groups of the form
𝐺
left-semidirect-product
𝐹
delimited-⟨⟩
𝑎
with the conditions
subscript
𝐶
𝐹
𝑎
1
and
𝑎
4
. The mapping
:
𝑎
→
𝐹
𝐹
defined by the rule
→
𝑡
superscript
𝑡
...𝑎
superscript
𝑎
1
𝑡
𝑎
is a fixed-point-free (regular) automorphism of the group
𝐹
. In this case, a finite group
𝐹
is solvable and its commutator subgroup is nilpotent (Gorenstein and Herstein, 1961), and a locally finite group
𝐹
is solvable and its second commutator subgroup is contained in the center
𝑍
𝐹
(Kovács, 1961). It is unknown whether a periodic group
𝐹
is always locally finite (Shumyatsky’s Question 12.100 from
The Kourovka Notebook
). We establish the following properties of groups. For
𝜋
𝜋
𝐹
𝜋
subscript
𝐶
𝐹
superscript
𝑎
2
, the group
𝐹
is
𝜋
-closed and the subgroup
subscript
𝑂
𝜋
𝐹
is abelian and is contained in
𝑍
superscript
𝑎
2
𝐹
(Theorem 1). A group
𝐹
without infinite elementary abelian
superscript
𝑎
2
-admissible subgroups is locally finite (Theorem 2). In a nonlocally finite group
𝐹
, there is a nonlocally finite
𝑎
-admissible subgroup factorizable by two locally finite
𝑎
-admissible subgroups (Theorem 3). For any positive integer
𝑛
divisible by an odd prime, we give examples of nonlocally finite periodic groups with a regular automorphism of order
𝑛
.
Self-complementary Cayley graphs are useful in the study of Ramsey numbers, but they are relatively very rare and hard to construct. In this paper, we construct several families of new ...self-complementary Cayley graphs of order $p^4$ where $p$ is a prime and congruent to $1$ modulo $8$.
Let G=Zp⊕Zp2, where p is a prime number. Suppose that d is a divisor of the order of G. In this paper, we find the number of automorphisms of G fixing d elements of G and denote it by θ(G,d). As a ...consequence, we prove a conjecture of Checco-Darling-Longfield-Wisdom. We also find the exact number of fixed-point-free automorphisms of the group Zpa⊕Zpb, where a and b are positive integers with a<b. Finally, we compute θ(D2q,d), where D2q is the dihedral group of order 2q, q is an odd prime, and d∈{1,q,2q}.
Let
C be a cyclic 2-group that acts fixed-point-freely on a group
K, and let
T
⊆
C
be the subgroup of index 2. The main result of this paper is that the square-free parts of the degrees of the
...T-invariant irreducible characters of
K are never divisible by primes
p
≡
−
1
mod
|
C
|
.
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