With this operation, we can first explain what a quotient group is, and then explain what a normal subgroup is:
Under this operation, the quotient group of group G is the division of its own group G..
It is completely determined by the subset containing e. A normal subgroup of G is a set containing E in any such partition. The subset in the partition is the coset of this normal subgroup.
Subgroup n of group G is a normal subgroup if and only if coset equation aN = Na holds for all A's in G, according to the binary operation on the subset defined above, the normal subgroup of G is a subgroup commutated with all subsets of G, which is expressed as n? G subgroups of all subgroups permuted in g are called permutation subgroups. Let n be a normal subgroup of group G, and we define the set G/N as the set of all left cosets of n in g, that is, G/N = {aN: a∈G}. The grouping operation on G/N is defined as above. In other words, for aN and bN in each G/N, the product of an and bN is (aN)(bN). This operation is closed because (aN)(bN) is actually a left coset:
(aN)(bN)= a(Nb)N = a(bN)N =(ab)NN =(ab)N .
The normality of n is used in this equation. Because of the normality of N, the left coset and right coset of N in G are equal, so G/N can also be defined as the set of all right cosets of N in G. Because the operation is derived from the product of subsets of G, it is well defined (independent of the specific choice of representation) and conforms to the associative law. The inverse of an element with unit element N. G/N is A? 1N .G/N is called quotient group, which comes from the division of integers. When we divide 12 by 3, we get the answer of 4, because we can recombine 12 objects into four subsets consisting of three objects. Quotient groups have the same idea, but use a group instead of a number as the final answer, because groups are more structured than randomly collected objects.
More specifically, when G/N is a normal subgroup of G, this group structure forms a natural "reorganization". They are cosets of n in G, because the final quotient we get from a group and normal subgroups contains more information than the number of cosets (generated by normal division), and here we get a group structure itself. Consider the group (under addition) of the integer set z and the subgroup 2Z composed of all even numbers. This is a normal subgroup, because Z is an Abelian group. There are only two cosets: an even set and an odd set; Therefore, the quotient group Z/2Z is a cyclic group of two elements. This quotient group is isomorphic to the group with the set {0, 1} of modular 2 addition operation; Unofficially, it is sometimes said that Z/2Z is equal to the set {0, 1} plus the module 2.
A summary of the last example. Then consider the addition group of integer set z. Let n be any positive integer. We consider the subgroup nZ of z composed of all multiples of n. NZ is still a normal subgroup in Z, because Z is an Abelian group. Coset collection {nZ, 1+nZ, ..., (n? 2)+nZ，(n？ 1)+nZ}. the integer k belongs to the coset r+nZ, where r is the remainder of k divided by n. The quotient Z/nZ can be regarded as the "residue" of a set of modulo n. This is a cyclic group of order n.
The coset of n in g considers the multiplicative Abel group g of the roots of a complex number of twelve units, which are points on the unit circle. They are shown as colored spheres on the right, and the amplitude of each point is marked with a number. Consider that its subgroup n consists of the quartic root of unit one, which is represented as a red ball in the figure. This normal subgroup decomposes the group into three cosets, which are represented as red, green and blue respectively. It can be verified that these cosets form a group of three elements (the product of red element and blue element is blue element, the inverse of blue element is green element, and so on). Therefore, the quotient group G/N is a group of three elements and a cyclic group of three elements.
Consider the group of real number set R under addition and subgroup Z of integer set. Cosets of z in R are all sets of form a+Z, where 0 ≤ A.
If G is a set of invertible 3 × 3 real matrices and N is a subgroup of 3 × 3 real matrices with determinant 1, then N is a normal subgroup in G (because it is the kernel of homomorphism of determinant). The coset of n is the set of matrices with a given determinant, so G/N is isomorphic to the multiplication group of non-zero real numbers.
Consider the Abelian group Z4 = Z/4Z (that is, the set with additive module 4 {0, 1 2,3}) and its subgroups {0,2}. The quotient group Z4/{0,2} is {{0,2}, {1, 3}}. This is a group identified as {0,2}, and the group operation is {0,2}+{1,3} = {1, 3}. Subgroup {0,2} and quotient group {{0,2}, {1, 3}} are isomorphic to Z2.
Consider multiplication groups. What is the nth remaining set n? Factorial method group. Then n is a normal subgroup in G and the factor group G/N has a coset n, (1+n) n, (1+n) 2n, …, (1+n) n? 1N .Pallier encryption system is based on the conjecture that it is difficult to determine the coset of random elements of g when the factorization of n is unknown. The quotient group G/G is isomorphic to a trivial group (a group with only one element), while G/G is isomorphic to G.
The order of G/N is defined as equal to [G: N], which is the exponent of the subgroup of N in G. If G is finite, this exponent is equal to the order of G divided by the order of N. Note that when both G and N are infinite (such as Z/2Z), G/N can be finite.
There is a "natural" morphism π: G → G/N, and the element G of each G is mapped to the coset of N to which G belongs, that is, π(g) = gN. The mapping π is sometimes called "gauge projection from g to G/N". Its core is n.
There is bijection mapping between the subgroup of G with N and the subgroup of G/N; If h is a subgroup of g containing n, then the subgroup of G/N is π(H). This mapping is also applicable to the normal subgroups of G and G/N, and is formalized in the lattice theorem.
Some important properties of quotient groups are recorded in homomorphism and isomorphism.
If G is an Abelian group, a nilpotent group or a solvable group, then so is G/n..
If g is a cyclic group or a finite generating group, so is g/n.
If n is contained in the center of G, then G is also called the central extension of this quotient group.
If H is a subgroup of a finite group G and its order is half that of G, then H is guaranteed to be a normal subgroup, so G/H exists and is isomorphic to C2. This result can also be expressed as "any subgroup with index 2 is a normal subgroup", which is also applicable to infinite groups.
All groups are isomorphic to the quotient of a free group.
Sometimes, but not necessarily, the group g can be reconstructed from G/N and n into a direct product or a semi-direct product. The problem of determining when it holds is called the inflation problem. An example of this is as follows. Z4/{0,2} is isomorphic to Z2 and isomorphic to {0,2}, but the only semi-direct product is the direct product, because Z2 has only a trivial automorphism. So Z4 is different from Z2 × Z2 and cannot be reconstructed. Quotient ring, also known as factor ring
Group expansion
Lattice theorem
Quotient category
Short exact sequence