Automorphisms of Finite Abelian Groups

@article{Hillar2007AutomorphismsOF,
  title={Automorphisms of Finite Abelian Groups},
  author={Christopher J. Hillar and Darren L. Rhea},
  journal={The American Mathematical Monthly},
  year={2007},
  volume={114},
  pages={917 - 923}
}
  • C. Hillar, D. Rhea
  • Published 8 May 2006
  • Mathematics, Computer Science
  • The American Mathematical Monthly
Much less known, however, is that there is a description of Aut(G), the automorphism group of G. The first compete characterization that we are aware of is contained in a paper by Ranum [1] near the turn of the last century. Beyond this, however, there are few other expositions [4]. Our goal is to fill this gap, thereby providing a much needed accessible and modern treatment. Our characterization of Aut(G) is accomplished in three main steps. The first observation is that it is enough to work… Expand
View on Taylor & Francis
msri.org
137 Citations
Highly Influential Citations
6
Background Citations
45
Methods Citations
12

137 Citations

Citation Type

Citation Type

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… Expand
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… Expand
On the Fixed Points of Abelian Group Automorphisms
In this article, we present general properties of xed-point groups of the automorphisms of nite groups. Speci cally, we determine the form of xed-point groups and partition Aut(G) according to the… Expand
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… Expand
Fuchs' problem for p-groups
Abstract Which groups can be the group of units in a ring? This open question, posed by Laszlo Fuchs in 1960, has been studied by the authors and others with a variety of restrictions on either the… Expand
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… Expand
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… Expand
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… Expand
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… Expand
ON THE CHARACTERISTIC DEGREE OF FINITE GROUPS
In this article we introduce and study the concept of characteristic degree of a subgroup in a finite group. We define the characteristic degree of a subgroup H in a finite group G as the ratio of… Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 16 REFERENCES
The order of the automorphism group of finite abelian group
Up to now,for any given finite abelian group,it havn't been found any paper gave a general formula to calculate the order of the automorphism group of the group.By discussing the bases of the group… Expand
Algebra 3rd ed
  • Algebra 3rd ed
  • 1993
An elementary derivation of the formulas of Wallis, Leibnitz and Euler for the number π (in Russian), Uspechi matematiceskich nauk
  • N. S.)
  • 1953
Wallis's og Brounckers formler for tc
  • Norsk matematisk tidskrift
  • 1951
Wallis's og Brounckers formler for π (in Norwegian)
  • Norsk matematisk tidskrift
  • 1951
674–686. Department of Mathematics, Texas A&M University, College Station, TX 77843 E-mail address: chillar@math.tamu.edu Department of Mathematics, University
  • drhea@math.berkeley.edu
  • 1928
ber die Automorphismen einer endlischen
  • Abelschen Gruppe, Math. Ann
  • 1928
Über die Automorphismen einer endlichen Abelschen Gruppe
λ + c0 be the characteristic polynomial of an n-by-n matrix A over a given field K. The elegant proof of the Cayley-Hamilton theorem of [1, p. 50] can be easily modified to prove that
  • College Station,
  • 1928
1 Aadj = An'x + cn_! An~2 +
  • + ci
...
1
2
...