The finite nonsolvable groups with at most 6 automorphism orbits are classified, there are only a finite number of them up to isomorphism. It is also proved that there are infinitely many finite non solvable groups with 7 automorphism orbits.
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 the finite non-solvable groups G with ω(G) ≤, 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. 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, generalizing a result of Kohl.
Finite groups with six or seven automorphism orbits
Garonzi M;
2017
Abstract
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 the finite non-solvable groups G with ω(G) ≤, 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. 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, generalizing a result of Kohl.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
