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.

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