# Hamiltonian group

A *Hamiltonian group ^{}* is a non-abelian group

^{}in which all subgroups

^{}(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 group^{} ${Q}_{8}$ of order $8$ (see the structure^{} 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 abelian^{} 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].

###### Theorem.

A group is Hamiltonian if and only if it is isomorphic^{} to ${Q}_{\mathrm{8}}\mathrm{\times}P$
for some periodic abelian group $P$ that has no element of order $\mathrm{4}$.

In particular, Hamiltonian groups are always periodic (in fact, locally finite^{}), 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 ${Q}_{8}\times B$, where $B$ is an elementary abelian $2$-group.

## References

- 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 |