Hamiltonian group
Encyclopedia
In group theory
, a Dedekind group is a group
G such that every subgroup
of G is normal
.
All abelian group
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
of order 8, denoted by Q8.
It can be shown that every Hamiltonian group is a direct product
of the form G = Q8 × B × D, where B is the direct sum of some number of copies of the cyclic group
C2, and D is a periodic
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
, the discoverer of quaternion
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 2a has 22a −6 quaternion groups as subgroups". In 2005 Horvat et al. used this structure to count the number of Hamiltonian groups of any order n = 2eo where o is an odd integer. When e ≤ 3 then there are no Hamiltonian groups of order n, otherwise there are the same number as there are Abelian groups of order o.
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a Dedekind group is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G such that every subgroup
Subgroup
In group theory, 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 if the restriction of * to H x H is a group operation on H...
of G is normal
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
.
All abelian group
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
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
Quaternion group
In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...
of order 8, denoted by Q8.
It can be shown that every Hamiltonian group is a direct product
Direct product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of the form G = Q8 × B × D, where B is the direct sum of some number of copies of the cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
C2, and D is a periodic
Periodic group
In group theory, a periodic group or a torsion group is a group in which each element has finite order. All finite groups are periodic. The concept of a periodic group should not be confused with that of a cyclic group, although all finite cyclic groups are periodic.The exponent of a periodic group...
abelian group with all elements of odd order.
Dedekind groups are named after Richard Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
, 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
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
, the discoverer of quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s.
In 1898 George Miller
George Abram Miller
George Abram Miller was an early group theorist whose many papers and texts were considered important by his contemporaries, but are now mostly considered only of historical importance...
delineated the structure of a Hamiltonian group in terms of its order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
and that of its subgroups. For instance, he shows "a Hamilton group of order 2a has 22a −6 quaternion groups as subgroups". In 2005 Horvat et al. used this structure to count the number of Hamiltonian groups of any order n = 2eo where o is an odd integer. When e ≤ 3 then there are no Hamiltonian groups of order n, otherwise there are the same number as there are Abelian groups of order o.