Hamiltonian group

In group theory , a group is called a Dedekind group (after Richard Dedekind ) if every subgroup is a normal divisor . Obviously every Abelian group is a Dedekindian group. The non-Abelian among them are called Hamiltonian groups (after William Rowan Hamilton ).

The Hamiltonian groups can be given in full according to a sentence that goes back to Dedekind:

• Every finite Hamiltonian group is of the form , where ${\displaystyle G}$ ${\displaystyle G\cong Q_{8}\times A\times (\mathbb {Z} /2\mathbb {Z} )^{n}}$
  • ${\displaystyle Q_{8}}$is the quaternion group ,
  • ${\displaystyle A}$an abelian group of odd -order is
  • and is. ${\displaystyle n\in \mathbb {N} _{0}}$

If the third factor is missing. The group can be one element , in which case the second factor is missing. The quaternion group is therefore the smallest Hamiltonian group and each Hamiltonian group contains a direct factor that is isomorphic to the quaternion group . ${\displaystyle n=0}$ ${\displaystyle A}$

Accordingly, and are not Hamiltonian groups. In fact, respectively, are or are non-normal subgroups, where as usual and is. ${\displaystyle Q_{8}\times Q_{8}}$ ${\displaystyle Q_{8}\times \mathbb {Z} /4\mathbb {Z} }$ ${\displaystyle \{(q,q);q\in Q_{8}\}}$ ${\displaystyle \{(1,{\overline {0}}),(i,{\overline {1}}),(-1,{\overline {2}}),(-i,{\overline {3}})\}\,}$ ${\displaystyle Q_{8}\,=\,\{1,-1,i,-i,j,-j,k,-k\}}$ ${\displaystyle \mathbb {Z} /4\mathbb {Z} =\{{\overline {0}},{\overline {1}},{\overline {2}},{\overline {3}}\}}$

Individual evidence

1. ↑ Bertram Huppert : Finite groups (= The basic teachings of the mathematical sciences in individual representations. With special consideration of the areas of application. Vol. 134). Volume 1. Springer, Berlin et al. 1967, sentence III, 7.12.

swell

• Richard Dedekind : About groups, all of whose members are normal members. In: Mathematical Annals . Vol. 84, No. 4, 1897, pp. 548-561, digitized .