mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H...

a direct product of objects already known, giving a new one. This induces a structure on the Cartesian product of the underlying sets from that of the...

specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect...

a product of groups usually refers to a direct product of groups, but may also mean: semidirect product Product of group subsets wreath product free...

product Cartesian product of sets Direct product of groups Semidirect product Product of group subsets Wreath product Free product Zappa–Szép product...

method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules...

Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects...

mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains...

means the n-fold direct product of groups. In general, a (possibly infinite) elementary abelian p-group is a direct sum of cyclic groups of order p. (Note...

dihedral groups, and the semidihedral groups. Furthermore, every finite nilpotent group is the direct product of p-groups. The multiplicative group of upper...

the direct sum is the same as the direct product. In the case of groups, if the group operation is written as + {\displaystyle +} the phrase "direct sum"...

category theory. Products over other kinds of algebraic structures include: the Cartesian product of sets the direct product of groups, and also the semidirect...

set of cohomology groups H n ( G , M ) {\displaystyle H^{n}(G,M)} . The cohomology groups in turn provide insight into the structure of the group G and...

category-theoretical product in Grp is just the direct product of groups while the category-theoretical coproduct in Grp is the free product of groups. The zero...

In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on...

mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly...

making it an example of a coproduct. Contrast with the direct product, which is the dual notion. The most familiar examples of this construction occur...

insurance products, including Direct Line and Churchill, Darwin as well as the roadside assistance and vehicle recovery provider Green Flag. The group is listed...

proof of Lemma 2.3: "the trivial group is the direct product of the empty family of groups" Baer, Reinhold (1937), "Abelian groups without elements of finite...

abelian group is a direct product of cyclic groups. Every cyclic group of prime order is a simple group, which cannot be broken down into smaller groups. In...

materials science. Group theory is also central to public key cryptography. Central extension Direct product of groups Direct sum of groups Extension problem...

substituent groups on the aromatic ring influence the overall reaction rate or have a directing effect on positional isomer of the products that are formed...

field of group theory, the central product is one way of producing a group from two smaller groups. The central product is similar to the direct product, but...

one can define a product of group subsets in a natural way. If S and T are subsets of a group G, then their product is the subset of G defined by S T...

groups: Any quotient of a cotorsion group is cotorsion. A direct product of groups is cotorsion if and only if each factor is. Every divisible group or...

(Unlike for the direct product of groups, the element of H may change if the order of the factors is changed.) 2.  Outer semidirect product: if N and H are...

the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps...

_{>0}} is a group, Q {\displaystyle Q} is a direct product of groups. The one-parameter subgroups L and H in Q profile the activity in the product, and L ×...

are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits...

number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the...