group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, a Dedekind group is a group ''G'' such that every

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...

of ''G'' is normal. All

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

s are Dedekind groups. A non-abelian Dedekind group is called a Hamiltonian group. The most familiar (and smallest) example of a Hamiltonian group is the

quaternion group In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a nonabelian group, non-abelian group (mathematics), group of Group order, order eight, isomorphic to the eight-element subset \ of the quaternions under multiplication. ...

of order 8, denoted by Q₈. Dedekind and Baer have shown (in the finite and respectively infinite order case) that every Hamiltonian group is a

direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...

of the form , where ''B'' is an

elementary abelian 2-group In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian group ...

, and ''D'' is a torsion abelian group with all elements of odd order. Dedekind groups are named after

Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...

, who investigated them in , proving a form of the above structure theorem (for finite groups). He named the non-abelian ones after

William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ire ...

, the discoverer of

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...

s. In 1898 George Miller delineated the structure of a Hamiltonian group in terms of its

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

and that of its subgroups. For instance, he shows "a Hamilton group of order 2^''a'' has quaternion groups as subgroups". In 2005 Horvat ''et al'' used this structure to count the number of Hamiltonian groups of any order where ''o'' is an odd integer. When then there are no Hamiltonian groups of order ''n'', otherwise there are the same number as there are Abelian groups of order ''o''.

Notes

References

* . * Baer, R. Situation der Untergruppen und Struktur der Gruppe, Sitz.-Ber. Heidelberg. Akad. Wiss.2, 12–17, 1933. * . * . *. *{{citation, first=Olga, last=Taussky, author-link=Olga Taussky-Todd, year=1970, title=Sums of squares, journal=

American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an ...

, volume= 77, issue=8, pages=805–830, mr=0268121, doi=10.2307/2317016, jstor=2317016, hdl=10338.dmlcz/120593, hdl-access=free. Group theory Properties of groups