Hamiltonian group

A Hamiltonian groupMathworldPlanetmath is a non-abelian groupMathworldPlanetmath in which all subgroupsMathworldPlanetmathPlanetmath (http://planetmath.org/Subgroup) are normal.

Richard Dedekind investigated finite Hamiltonian groups in 1895, and proved that they all contain a copy of the quaternion groupMathworldPlanetmathPlanetmath Q8 of order 8 (see the structureMathworldPlanetmath theorem below). He named them in honour of William Hamilton, the discoverer of quaternions.

Groups in which all subgroups are normal (that is, groups that are either abelianMathworldPlanetmath or Hamiltonian) are sometimes called Dedekind groups, or quasi-Hamiltonian groups.

The following structure theorem was proved in its full form by Baer[1], but Dedekind already came close to it in his original paper[2].


A group is Hamiltonian if and only if it is isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to Q8×P for some periodic abelian group P that has no element of order 4.

In particular, Hamiltonian groups are always periodic (in fact, locally finitePlanetmathPlanetmathPlanetmath), nilpotent of class 2, and solvable of length 2.

From the structure theorem one can also see that the only Hamiltonian p-groups (http://planetmath.org/PGroup4) are 2-groups of the form Q8×B, where B is an elementary abelian 2-group.


  • 1 R. Baer, Situation der Untergruppen und Struktur der Gruppe, S. B. Heidelberg. Akad. Wiss. 2 (1933), 12–17.
  • 2 R. Dedekind, Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind, Mathematische Annalen 48 (1897), 548–561. (This paper is http://gdz.sub.uni-goettingen.de/dms/resolveppn/?GDZPPN002256258available from GDZ.)
Title Hamiltonian group
Canonical name HamiltonianGroup
Date of creation 2013-03-22 15:36:14
Last modified on 2013-03-22 15:36:14
Owner yark (2760)
Last modified by yark (2760)
Numerical id 15
Author yark (2760)
Entry type Definition
Classification msc 20F24
Classification msc 20F18
Classification msc 20F50
Synonym Hamilton group
Defines Dedekind group
Defines quasi-Hamiltonian group
Defines Hamiltonian