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 ide ...
''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 subgroup ...
of ''G'' is
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
.
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 commut ...
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 non-abelian group of order eight, isomorphic to the eight-element subset
\ of the quaternions under multiplication. It is given by the group presentation
:\mathrm_8 ...
of order 8, denoted by Q
8.
Dedekind and
Baer
Baer (or Bär, from german: bear, links=no) or Van Baer is a surname. Notable people with the surname include:
Baer
* Alan Baer, American tuba player
* Arthur "Bugs" Baer (1886–1969), American journalist and humorist
* Buddy Baer (1915–198 ...
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 ta ...
of the form , where ''B'' is an
elementary abelian 2-group, and ''D'' is a
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
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 group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
s). He named the non-abelian ones after
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
, 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 quatern ...
s.
In 1898
George Miller delineated the structure of a Hamiltonian group in terms of its
order 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 e ...
, 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