Soluble groups with few orbits under automorphisms

Journal of Algebra, 2018

Let G be a group. The orbits of the natural action of Aut(G) on G are called "automorphism orbits" of G, and the number of automorphism orbits of G is denoted by ω(G). In this paper we prove that if G is an FC-group with finitely many automorphism orbits, then the derived subgroup G ′ is finite and G admits a decomposition G = Tor(G)× D, where Tor(G) is the torsion subgroup of G and D is a divisible characteristic subgroup of Z(G). We also show that if G is an infinite FC-group with ω(G) 8, then either G is soluble or G ∼ = A 5 ×H, where H is an infinite abelian group with ω(H) = 2. Moreover, we describe the structure of the infinite non-soluble FC-groups with at most eleven automorphism orbits.

Journal of Group Theory, 2017

Let G be a group. The orbits of the natural action of Aut ⁡ ( G ) {\operatorname{Aut}(G)} on G are called “automorphism orbits” of G, and the number of automorphism orbits of G is denoted by ω ⁢ ( G ) {\omega(G)} . In this paper the finite non-solvable groups G with ω ⁢ ( G ) ≤ 6 {\omega(G)\leq 6} are classified – this solves a problem posed by Markus Stroppel – and it is proved that there are infinitely many finite non-solvable groups G with ω ⁢ ( G ) = 7 {\omega(G)=7} . Moreover, it is proved that for a given number n there are only finitely many finite groups G without non-trivial abelian normal subgroups and such that ω ⁢ ( G ) ≤ n {\omega(G)\leq n} , generalizing a result of Kohl.

2007

Among other things, we prove that the group of automorphisms fixing every normal subgroup of a nilpotent-by-abelian group is nilpotent-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian group is soluble of derived length at most 3. An example shows that this bound cannot be improved.

Proceedings - Mathematical Sciences, 2018

The aim of this paper is to state some results on an α-nilpotent group, which was recently introduced by Barzegar and Erfanian (Caspian J. Math. Sci. 4(2) (2015) 271-283), for any fixed automorphism α of a group G. We define an identity nilpotent group and classify all finitely generated identity nilpotent groups. Moreover, we prove a theorem on a generalization of the converse of the known Schur's theorem. In the last section of the paper, we study absolute normal subgroups of a finite group.

Journal of the Australian Mathematical Society, 1986

Let G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

Journal of Algebra, 2000

We show that certain properties of groups of automorphisms can be read off from the actions they induce on the finite characteristic quotients of their underlying group G. In particular, we obtain criteria for groups of automorphisms of a Ž. residually finite and soluble minimax-by-finite group G to be nilpotent or soluble. Ž. Moreover, we give explicit bounds on the class the derived length, resp. of such groups of automorphisms in terms of invariants of G. Finally, we consider similar questions when G is the free group of rank two.

Monatshefte für Mathematik, 2016

A finite group F H 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. Let F H be a Frobenius-like group with complement H of prime order such that C F (H) is of prime order. Suppose that F H acts on a finite group G by automorphisms where (|G|, |H |) = 1 in such a way that C G (F) = 1. In the present paper we prove that the Fitting series of C G (H) coincides with the intersections of C G (H) with the Fitting series of G, and the nilpotent length of G exceeds the nilpotent length of C G (H) by at most one. As a corollary, we also prove that for any set of primes π , the upper π-series of C G (H) coincides with the intersections of C G (H) with the upper π-series of G, and the π-length of G exceeds the π-length of C G (H) by at most one. Keywords Frobenius-like group • Fixed points • Nilpotent length • π-length Mathematics Subject Classification 20D10 • 20D15 • 20D45 Communicated by J. S. Wilson.

Proceedings of The National Academy of Sciences, 1959

Let G be a finite group. We let m(G) and σ(G) denote the number of maximal subgroups of G and the least positive integer n such that G is written as the union of n proper subgroups, respectively. In this paper, we determine the structure of G/Φ(G) when G is a finite soluble group with m(G) ≤ 2σ(G).

Bulletin of the Australian Mathematical Society, 1969

In a finite soluble group G, the Fitting (or nilpotency) length h(G) can be considered as a measure for how strongly G deviates from being nilpotent. As another measure for this, the number v(G) of conjugacy classes of the maximal nilpotent subgroups of G may be taken. It is shown that there exists an integer-valued function f on the set of positive integers such that h(G) ≦ f(v(G)) for all finite (soluble) groups of odd order. Moreover, if all prime divisors of the order of G are greater than v(G)(v(G) - l)/2, then h(G) ≦3. The bound f(v(G)) is just of qualitative nature and by far not best possible. For v(G) = 2, h(G) = 3, some statements are made about the structure of G.