Group obtained by aggregating similar elements of a larger group
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A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory.
For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written , where is the original group and is the normal subgroup. (This is pronounced , where is short for modulo.)
Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of . Specifically, the image of under a homomorphism is isomorphic to where denotes the kernel of .
The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects.
For other examples of quotient objects, see quotient ring, quotient space (linear algebra), quotient space (topology), and quotient set.
A quotientgroup or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that...
arithmetic, a quotient (from Latin: quotiens 'how many times', pronounced /ˈkwoʊʃənt/) is a quantity produced by the division of two numbers. The quotient has widespread...
a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotientgroup in group theory...
alternative generator of G. Instead of the quotient notations Z/nZ, Z/(n), or Z/n, some authors denote a finite cyclic group as Zn, but this clashes with the notation...
quotient spaces in linear algebra, quotient spaces in topology, quotientgroups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories...
The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotientgroups, and covering groups. These are listed...
gives rise to a quotientgroup. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the...
that any topological group can be canonically associated with a Hausdorff topological group by taking an appropriate canonical quotient; this however, often...
space (that is, a quotient ring is the quotient of a ring by an ideal, not a subring, and a quotientgroup is the quotient of a group by a normal subgroup...
G is called solvable if it has a subnormal series whose factor groups (quotientgroups) are all abelian, that is, if there are subgroups 1 = G 0 ◃ G 1...
construct quotientgroups of the given group. Furthermore, the normal subgroups of G {\displaystyle G} are precisely the kernels of group homomorphisms...
describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie...
the other direction, if G is any topological group and K is a discrete normal subgroup of G then the quotient map p : G → G / K is a covering homomorphism...
isomorphism theorem states that the image of a group homomorphism, h(G) is isomorphic to the quotientgroup G/ker h. The kernel of h is a normal subgroup...
theorem. The quotient of a Lie group by a closed normal subgroup is a Lie group. The universal cover of a connected Lie group is a Lie group. For example...
inverse, which, in this ring, are exactly those coprime to n. This quotientgroup, usually denoted ( Z / n Z ) × {\displaystyle (\mathbb {Z} /n\mathbb...
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotientgroup. If Q {\displaystyle...
of abstract groups is given by the construction of a factor group, or quotientgroup, G/H, of a group G by a normal subgroup H. Class groups of algebraic...
such that the quotientgroup G/A is abelian. Subgroups of metabelian groups are metabelian, as are images of metabelian groups over group homomorphisms...
smaller groups, namely a nontrivial normal subgroup and the corresponding quotientgroup. This process can be repeated, and for finite groups one eventually...
projective special linear group PSL(2, Z), which is the quotient of the 2-dimensional special linear group SL(2, Z) over the integers by its center {I, −I}....
quotientgroup is itself a Coxeter group, and the Coxeter graph of the affine Coxeter group is obtained from the Coxeter graph of the quotientgroup by...
in the category of (locally small) categories, analogous to a quotientgroup or quotient space, but in the categorical setting. Let C be a category. A...
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