The quotient group Aut ( G ) / Inn ( G ) is usually denoted by Out ( G ); the non-trivial elements are the cosets that contain the outer automorphisms.

The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian.

The quotient G / is an abelian group called the abelianization of G or G made abelian.

He found that an equation could be solved in radicals if one can find a series of subgroups of its Galois group, each one normal in its successor with abelian quotient, or its Galois group is solvable.

In mathematics, specifically group theory, a quotient group ( or factor group ) is a group obtained by identifying together elements of a larger group using an equivalence relation.

The resulting quotient is written, where G is the original group and N is the normal subgroup.

The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism is isomorphic to where ker ( φ ) denotes the kernel of φ.

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.

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

He calculated what he called a Intelligenz-Quotient score, or IQ, as the quotient of the ' mental age ' ( the age group which scored such a result on average ) of the test-taker and the ' chronological age ' of the test-taker, multiplied by 100.

* The quotient of a Lie group by a closed normal subgroup is a Lie group.

The identity component of any Lie group is an open normal subgroup, and the quotient group is a discrete group.

The set of all possible equivalence classes of X by ~, denoted, is the quotient set of X by ~.

The set of all equivalence classes in given an equivalence relation is denoted as and called the quotient set of by.

This operation can be thought of ( very informally ) as the act of dividing the input set by the equivalence relation, hence both the name " quotient ", and the notation, which are both reminiscent of division.

One way in which the quotient set resembles division is that if is finite and the equivalence classes are all equinumerous, then the number of equivalence classes in can be calculated by dividing the number of elements in by the number of elements in each equivalence class.

Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set of all equivalence classes of ~.

where v, v < sub > 1 </ sub > and v < sub > 2 </ sub > are vectors from V, while w, w < sub > 1 </ sub >, and w < sub > 2 </ sub > are vectors from W, and c is from the underlying field K. Denoting by R the space generated by these four equivalence relations, the tensor product of the two vector spaces V and W is then the quotient space

Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes ( or congruence classes ) for the relation.

Together, these equivalence classes are the elements of a quotient group.

For a congruence on a ring, the equivalence class containing 0 is always a two-sided ideal, and the two operations on the set of equivalence classes define the corresponding quotient ring.

An algebraic structure in a variety may be understood as the quotient algebra of term algebra ( also called " absolutely free algebra ") divided by the equivalence relations generated by a set of identities.

These equations induce equivalence classes on the free algebra ; the quotient algebra then has the algebraic structure of a group.

This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be the quotient set N² /~, i. e. we identify two pairs ( a, b ) and ( c, d ) if they are equivalent in the above sense.

If X is a diffeological space and ~ is some equivalence relation on X, then the quotient set X /~ has the diffeology generated by all compositions of plots of X with the projection from X to X /~.

Consider the polynomial ring R, and the irreducible polynomial The quotient space is given by the congruence As a result, the elements ( or equivalence classes ) of are of the form where a and b belong to R. To see this, note that since it follows that,,, etc.

Then the quotient sheaf consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms.

In mathematics, a quotient algebra, ( where algebra is used in the sense of universal algebra ), also called a factor algebra is obtained by partitioning the elements of an algebra in equivalence classes given by a congruence, that is an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense described below.

In terms of groups, the residue class is the coset of a in the quotient group, a cyclic group.

The creative class quotient for Portales was 21 % in 2007.

That is, the residue class is the coset of n in the quotient ring Z / kZ.

In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra.

Class field theory also includes a reciprocity homomorphism which acts from the idele class group of a global field, i. e. the quotient of the ideles by the multiplicative group of the field, to the Galois group of the maximal abelian extension of the global field.

of the maximal abelian quotient of the Galois group of the extension with the quotient of the idele class group of K by the image of the norm of the idele class group of L.

The quotient of its multiplicative group by the multiplicative group of the algebraic number field is the central object in class field theory.

The principal idèles are given by the diagonal embedding of the invertible elements of the number field or field of functions and the quotient of the idele group by principal ideles is the idele class group.

For an algebra, given a congruence E on, the algebra is called the quotient algebra ( or factor algebra ) of modulo E. There is a natural homomorphism from to mapping every element to its equivalence class.

However, if sets whose symmetric difference has measure zero are identified into a single equivalence class, the resulting quotient set can be properly metrized by the induced metric.

Given that we are only interested in what happens on shell, we would often take the quotient by the ideal generated by the Euler-Lagrange equations, or in other words, consider the equivalence class of functionals / flows which agree on shell.

) For any group H lying between I < sub > m </ sub > and P < sub > m </ sub >, the quotient I < sub > m </ sub >/ H is called a generalized ideal class group.

In geometric topology especially, one considers the quotient group obtained by quotienting out by isotopy, called the mapping class group:

The quotient group is called the mapping class group of X.

The narrow class group is defined to be the quotient

As | p > is really an element of the BRST cohomology, i. e. a quotient space, it is really an equivalence class of states.