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 comm ...
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 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, 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 Irela ...
, the discoverer of
quaternions.
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, 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