In
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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
''G'' such that every
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
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 commu ...
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 Q
8.
Dedekind and
Baer have shown (in the finite and respectively infinite order case) that every Hamiltonian group is a
direct product
In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...
of the form , where ''B'' is an
elementary abelian 2-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. H ...
, who investigated them in , proving a form of the above structure theorem (for
finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
s). He named the non-abelian ones after
William Rowan Hamilton
Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...
, 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. The algebra of quater ...
s.
In 1898
George Miller delineated the structure of a Hamiltonian group in terms of its
order
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
...
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 peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...
, 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