İsmail Şuayip Güloğlu

Finite groups admitting fixed-point free automorphisms of order pqr

Let G be a finite group and A be a group of operators of G with CGðAÞ 1⁄4 1. In [14], Turull proved that if ðjGj; jAjÞ 1⁄4 1 then (with certain exceptions for A), the Fitting height of G is bounded by the length of the longest chain of subgroups of A. We expect a similar bound for the Fitting height of G when the assumption ðjGj; jAjÞ 1⁄4 1 is replaced by the assumption that A is nilpotent (see [1]). In [9], Cheng Kei-Nah showed that G is metanilpotent if A is a cyclic group whose order is a product of two distinct primes. Here we obtain a result that takes Kei-Nah’s work one step further:

Action of a Frobenius-like group with fixed-point free kernel

Abstract We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that [F,h]=F

Nilpotent Length of a Finite Solvable Group with a Frobenius Group of Automorphisms

{[ F,h] =F}

Action of a frobenius-like group with kernel having central derived subgroup

for all nonidentity elements h ∈ H. We prove that any irreducible nontrivial FH-module for a Frobenius-like group FH of odd order over an algebraically-closed field has an H-regular direct summand if either F is fixed-point free on V or F acts nontrivially on V and the characteristic of the field is coprime to the order of F. Some consequences of this result are also derived.

A combinatorial discussion on finite dimensional leavitt path algebras

We prove that a finite solvable group G admitting a Frobenius group FH of automorphisms of coprime order with kernel F and complement H such that [G, F] = G and C C G (F)(h) = 1 for all nonidentity elements h ∈ H, is of nilpotent length equal to the nilpotent length of the subgroup of fixed points of H.

Fixed-point free action of an abelian group of odd non-squarefree exponent

A finite group FH is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that [F,h] = F for all nonidentity elements h ∈ H. Suppose that a finite group G admits a Frobenius-like group of automorphisms FH of coprime order with [F′,H] = 1. In case where CG(F) = 1 we prove that the groups G and CG(H) have the same nilpotent length under certain additional assumptions.

Finite groups whose character degree graphs coincide with their prime graphs

Any finite dimensional semisimple algebra A over a field K is isomorphic to a direct sum of finite dimensional full matrix rings over suitable division rings. We shall consider the direct sum of finite dimensional full matrix rings over a field K. All such finite dimensional semisimple algebras arise as finite dimensional Leavitt path algebras. For this specific finite dimensional semisimple algebra A over a field K, we define a uniquely determined specific graph - called a truncated tree associated with A - whose Leavitt path algebra is isomorphic to A. We define an algebraic invariant κ(A) for A and count the number of isomorphism classes of Leavitt path algebras with the same fixed value of κ(A). Moreover, we find the maximum and the minimum K-dimensions of the Leavitt path algebras of possible trees with a given number of vertices and we also determine the number of distinct Leavitt path algebras of line graphs with a given number of vertices.

The Influence of Hughes Type Action

Let A be a finite group acting fixed-point freely on a finite (solvable) group G. A longstanding conjecture is that if (|G|, |A|) = 1, then the Fitting length of G is bounded by the length of the longest chain of subgroups of A. It is expected that the conjecture is true when the coprimeness condition is replaced by the assumption that A is nilpotent. We establish the conjecture without the coprimeness condition in the case where A is an abelian group whose order is a product of three odd primes and where the Sylow 2-subgroups of G are abelian.

Rank and Order of a Finite Group Admitting a Frobenius-Like Group of Automorphisms

In the literature, there are several graphs related to a finite group G. Two of them are the character degree graph, denoted by ΔG), and the prime graph ΓG), In this paper we classify all finite groups whose character degree graphs are disconnected and coincide with their prime graphs. As a corollary, we find all finite groups whose character degree graphs are square and coincide with their prime graphs.

Fixed point free action on groups of odd order

We call the action of an automorphism α of a finite group G a Hughes type action if it is described by conditions on the orders of elements of G ⟨ α ⟩ − G. In the present paper we study the structure of finite group G admitting an automorphism α of prime order p so that the orders of elements in G ⟨ α ⟩ − G are not divisible by p 2.