In group theory, a Dedekind group is a group G such that every subgroup of G is normal.
All abelian groups are Dedekind groups.
A non-abelian Dedekind group is called a Hamiltonian group.[1]
The most familiar (and smallest) example of a Hamiltonian group is the quaternion group 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 of the form G = Q8 × B × D, 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, who investigated them in (Dedekind 1897), proving a form of the above structure theorem (for finite groups). He named the non-abelian ones after William Rowan Hamilton, 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 2a has 22a − 6 quaternion groups as subgroups". In 2005 Horvat et al[2] 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.
^Hall (1999). The theory of groups. p. 190.
^Horvat, Boris; Jaklič, Gašper; Pisanski, Tomaž (2005-03-09). "On the Number of Hamiltonian Groups". arXiv:math/0503183.
group theory, a Dedekindgroup is a group G such that every subgroup of G is normal. All abelian groups are Dedekindgroups. A non-abelian Dedekind group...
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors...
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example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain. Ideal class groups (or, rather, what...
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mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function D of three integer variables. Dedekind introduced...
credited to Richard Dedekind, in reference to (Dedekind 1877). The map of groups (2, 3, ∞) → (2, 3, n) (from modular group to triangle group) can be visualized...
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which...
mathematical logic, the Peano axioms (/piˈɑːnoʊ/, [peˈaːno]), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers...
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context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like...
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the lattice of subgroups of a group, and is sometimes called the Jordan–Dedekind chain condition. Moreover, a finite group is supersolvable if and only...
cannot carry a group structure is from sets X {\displaystyle X} with the following two properties: X {\displaystyle X} is an infinite Dedekind-finite set...
group Composition series Nilpotent group Divisible groupDedekindgroup, Hamiltonian group Examples Examples of groups Trivial group Additive group Permutation...
embedding of Q8 in the symmetric group S8, in addition to the embeddings given by the regular representations. Richard Dedekind considered the field Q [ 2 ...
semimodularity. Modular lattices are sometimes called Dedekind lattices after Richard Dedekind, who discovered the modular identity in several motivating...
single object, the idele class group, that describes both the quotient by this lattice and the ideal class group. The Dedekind zeta function of a number field...
R modules coincide with the divisible R modules if and only if R is a Dedekind domain. Injective object Injective module Griffith, p.6 Hall, p.197 Griffith...
also completed the Jordan–Hölder theorem. Dedekind and Miller independently characterized Hamiltonian groups and introduced the notion of the commutator...
^{2}+1240002q^{3}+10698752q^{4}+\cdots \end{aligned}}} and η(τ) is the Dedekind eta function. Wilson (1999) found the 30 conjugacy classes of maximal subgroups...
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they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domains...
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