Search results
Results from the WOW.Com Content Network
The quotient group is the same idea, although one ends up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects: in the quotient / , the group structure is used to form a natural "regrouping".
This has the intuitive meaning that the images of x and y are supposed to be equal in the quotient group. Thus, for example, r n in the list of relators is equivalent with =. [1] For a finite group G, it is possible to build a presentation of G from the group multiplication table, as follows.
quotient group Given a group G and a normal subgroup N of G, the quotient group is the set G / N of left cosets {aN : a ∈ G} together with the operation aN • bN = abN. The relationship between normal subgroups, homomorphisms, and factor groups is summed up in the fundamental theorem on homomorphisms.
Let T be a spanning tree for Y and define the fundamental group Γ to be the group generated by the vertex groups G x and elements y for each edge of Y with the following relations: y = y −1 if y is the edge y with the reverse orientation. y φ y,0 (x) y −1 = φ y,1 (x) for all x in G y. y = 1 if y is an edge in T. This definition is ...
A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two (usually independent ) random variables X and Y , the distribution of the random variable Z that is formed as the ratio Z = X / Y is a ratio ...
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group .
If a group H is a homomorphic image (or a quotient group) of a group G then rank(H) ≤ rank(G). If G is a finite non-abelian simple group (e.g. G = A n, the alternating group, for n > 4) then rank(G) = 2. This fact is a consequence of the Classification of finite simple groups.
In the category of sets, the coequalizer of two functions f, g : X → Y is the quotient of Y by the smallest equivalence relation ~ such that for every x ∈ X, we have f(x) ~ g(x). [1] In particular, if R is an equivalence relation on a set Y, and r 1, r 2 are the natural projections (R ⊂ Y × Y) → Y then the coequalizer of r 1 and r 2 is ...