Nilpotent length of a finite group admitting a frobenius group of automorphisms with fixed-point-free kernel

2011 ◽  
Vol 49 (6) ◽  
pp. 551-560 ◽  
Author(s):  
E. I. Khukhro
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Frobenius Group ◽  
Group Of Automorphisms ◽  
Nilpotent Length ◽  
A Finite Group

scholarly journals Action of a Frobenius-like group with kernel having central derived subgroup

2016 ◽  
Vol 26 (06) ◽  
pp. 1257-1265 ◽  
Author(s):  
Gülin Ercan ◽  
İsmail Ş. Güloğlu
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Group Of Automorphisms ◽  
Nilpotent Length ◽  
Derived Subgroup ◽  
A Finite Group ◽  
Coprime Order ◽  
Additional Assumptions

A finite group [Formula: see text] is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup [Formula: see text] with a nontrivial complement [Formula: see text] such that [Formula: see text] for all nonidentity elements [Formula: see text]. Suppose that a finite group [Formula: see text] admits a Frobenius-like group of automorphisms [Formula: see text] of coprime order with [Formula: see text] In case where [Formula: see text] we prove that the groups [Formula: see text] and [Formula: see text] have the same nilpotent length under certain additional assumptions.

A Sufficient Condition for Solvability in Groups Admitting Elementary Abelian Operator Groups

1977 ◽  
Vol 29 (4) ◽  
pp. 848-855 ◽  
Author(s):  
Martin R. Pettet
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Abelian Group ◽  
Prime Divisor ◽  
Sylow Subgroup ◽  
Elementary Abelian Group ◽  
Sufficient Condition ◽  
Group Of Automorphisms ◽  
Critical Observation ◽  
A Finite Group

Generalizing a celebrated theorem of Thompson, R. P. Martineau has established [4; 5] that a finite group which admits an elementary abelian group of automorphisms with trivial fixed-point subgroup is necessarily solvable. A critical observation in his approach to this problem is the fact that, corresponding to each prime divisor of its order, such a group contains a unique Sylow subgroup invariant (as a set) under the action. Hence, the theorem we shall derive here represents a modest extension of Martineau's result.

Half-Transitive Automorphism Groups

1966 ◽  
Vol 18 ◽  
pp. 1243-1250 ◽  
Author(s):  
I. M. Isaacs ◽  
D. S. Passman
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Permutation Group ◽  
Automorphism Groups ◽  
Permutation Representation ◽  
Group Of Automorphisms ◽  
Transitive Automorphism ◽  
A Finite Group ◽  
Special Case ◽  
Identity Elements

Let G be a finite group and A a group of automorphisms of G. Clearly A acts as a permutation group on G#, the set of non-identity elements of G. We assume that this permutation representation is half transitive, that is all the orbits have the same size. A special case of this occurs when A acts fixed point free on G. In this paper we study the remaining or non-fixed point free cases. We show first that G must be an elementary abelian g-group for some prime q and that A acts irreducibly on G. Then we classify all such occurrences in which A is a p-group.

A note on Frobenius–Wielandt groups

2019 ◽  
Vol 22 (4) ◽  
pp. 637-645
Author(s):  
Gil Kaplan
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Frobenius Group ◽  
Point Of View ◽  
Coset Space ◽  
Frobenius Groups ◽  
Frobenius Theorem ◽  
Frobenius Kernel ◽  
Frobenius Complement ◽  
A Finite Group

AbstractLet G be a finite group. G is called a Frobenius–Wielandt group if there exists {H<G} such that {U=\langle H\cap H^{g}\mid g\in G-H\rangle} is a proper subgroup of H. The Wielandt theorem [H. Wielandt, Über die Existenz von Normalteilern in endlichen Gruppen, Math. Nachr. 18 1958, 274–280; Mathematische Werke Vol. 1, 769–775] on the structure of G generalizes the celebrated Frobenius theorem. From a permutation group point of view, considering the action of G on the coset space {G/H}, it states in particular that the subgroup {D=D_{G}(H)} generated by all derangements (fixed-point-free elements) is a proper subgroup of G. Let {W=U^{G}}, the normal closure of U in G. Then W is the subgroup generated by all elements fixing at least two points. We present the proof of the Wielandt theorem in a new way (Theorem 1.6, Corollary 1.7, Theorem 1.8) such that the unique component whose proof is not elementary or by the Frobenius theorem is the equality {W\cap H=U}. This presentation shows what can be achieved by elementary arguments and how Frobenius groups are involved in one case of Frobenius–Wielandt groups. To be more precise, Theorem 1.6 shows that there are two possible cases for a Frobenius–Wielandt group G with {H<G}: (a) {W=D} and {G=HW}, or (b) {W<D} and {HW<G}. In the latter case, {G/W} is a Frobenius group with a Frobenius complement {HW/W} and Frobenius kernel {D/W}.

A GENERALIZED FIXED POINT FREE AUTOMORPHISM OF PRIME POWER ORDER

2012 ◽  
Vol 22 (04) ◽  
pp. 1250029
Author(s):  
GÜLİN ERCAN ◽  
İSMAİL Ş. GÜLOĞLU
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Nilpotent Group ◽  
Prime Power ◽  
Prime Power Order ◽  
Nilpotent Length ◽  
A Finite Group

Let G be a finite group and α be an automorphism of G of order pn for an odd prime p. Suppose that α acts fixed point freely on every α-invariant p′-section of G, and acts trivially or exceptionally on every elementary abelian α-invariant p-section of G. It is proved that G is a solvable p-nilpotent group of nilpotent length at most n + 1, and this bound is best possible.

scholarly journals A note on the fixed subring of an FPF ring

1989 ◽  
Vol 40 (1) ◽  
pp. 109-111 ◽  
Author(s):  
John Clark
Keyword(s):  
Finite Group ◽  
Associative Ring ◽  
Finitely Generated ◽  
Group Of Automorphisms ◽  
Direct Sum ◽  
Epimorphic Image ◽  
A Finite Group

An associative ring R with identity is called a left (right) FPF ring if given any finitely generated faithful left (right) R-module A and any left (right) R-module M then M is the epimorphic image of a direct sum of copies of A. Faith and Page have asked if the subring of elements fixed by a finite group of automorphisms of an FPF ring need also be FPF. Here we present examples showing the answer to be negative in general.

Nilpotent Partition-Inducing Automorphism Groups

1981 ◽  
Vol 33 (2) ◽  
pp. 412-420 ◽  
Author(s):  
Martin R. Pettet
Keyword(s):  
Finite Group ◽  
Automorphism Groups ◽  
Group Of Automorphisms ◽  
A Finite Group ◽  
Identity Elements

If A is a group acting on a set X and x ∈ X, we denote the stabilizer of x in A by CA(x) and let Γ(x) be the set of elements of X fixed by CA(x). We shall say the action of A is partitive if the distinct subsets Γ(x), x ∈ X, partition X. A special example of this phenomenon is the case of a semiregular action (when CA (x) = 1 for every x ∈ X so the induced partition is a trivial one). Our concern here is with the case that A is a group of automorphisms of a finite group G and X = G#, the set of non-identity elements of G. We shall prove that if A is nilpotent, then except in a very restricted situation, partitivity implies semiregularity.

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