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On finite groups having an automorphism with a large cycle

We characterize the finite groups with an automorphism permuting at least half of the group’s elements in one cycle. On the way, we provide an upper bound for the orders of elements of GLn(Z/p Z),…

AUTOMORPHISMS OF THE UNIT GROUPS OF SQUARE RADICAL ZERO FINITE COMMUTATIVE COMPLETELY PRIMARY RINGS

Let G be a group. The groups G ' for which G is an automorphism group have not been fully characterized. Suppose R is a Completely Primary finite Ring with Jacobson Radical J such that J 2 = (0). In…

Automorphism Groups of Quandles

We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In…

On The Determination of Sets By Their Subset Sums

. Let A be a multiset with elements in an abelian group. Let FS( A ) be the multiset containing the 2 | A | sums of all subsets of A . We study the reconstruction problem “Given FS( A ), is it…

Nilpotent groups related to an automorphism

AbstractThe aim of this paper is to state some results on an $$\alpha $$α-nilpotent group, which was recently introduced by Barzegar and Erfanian (Caspian J. Math. Sci. 4(2) (2015) 271–283), for any…

General Linear Groups as Automorphism Groups

Let G be a finite p-group, where p is a prime number and Aut(G )b e the automorphism group of G. We prove that if Aut(G) is isomorphic to GL(n,p) for some positive integer n, then G is an elementary…

Fixed points of automorphisms of certain finite groups

Let G = Zp2 × Zp3 be a finite group of order p5. Suppose that d is a divisor of the order of group G. In this article, we enumerated all automorphisms of G fixing d elements of G and denote them by…

Abelian Hopf Galois structures on prime-power Galois field extensions

The main theorem of this paper is that if (N, +) is a finite abelian p-group of p-rank m where m + 1 < p, then every regular abelian subgroup of the holomorph of N is isomorphic to N . The proof…

Finite groups with only small automorphism orbits

Abstract We study finite groups G such that the maximum length of an orbit of the natural action of the automorphism group Aut ⁢ ( G ) {\mathrm{Aut}(G)} on G is bounded from above by a constant. Our…