# Factor group

The factor group or quotient group is a group that is formed from a given group by means of a standard construction with the help of a normal divisor . It is denoted by and is the set of secondary classes . ${\ displaystyle G}$ ${\ displaystyle N \ trianglelefteq G}$${\ displaystyle G / N}$

## construction

The elements of are related to the minor classes , so ${\ displaystyle G / N}$${\ displaystyle N}$

${\ displaystyle G / N: = \ {gN: g \ in G \}}$.

The inner join is defined as ${\ displaystyle \ circ \ colon G / N \ times G / N \ rightarrow G / N}$

${\ displaystyle (gN) \ circ (hN): = (gh) N}$.

One can show with the help of the normal part property of that this link is well defined and that it is a group. This group is called the factor group from to . The neutral element of is and the inverse element of is given by. ${\ displaystyle N}$${\ displaystyle (G / N, \ circ)}$${\ displaystyle G}$${\ displaystyle N}$${\ displaystyle G / N}$${\ displaystyle N}$${\ displaystyle gN}$${\ displaystyle g ^ {- 1} N}$

The product is the same as the complex product . Conversely, one can show that a subgroup of a group is a normal subgroup if the equality holds for all . ${\ displaystyle (gN) \ circ (hN) = (gh) N}$ ${\ displaystyle (gN) \ cdot (hN)}$${\ displaystyle U}$${\ displaystyle (G, \ cdot)}$${\ displaystyle g, h \ in G}$${\ displaystyle (PDO) \ cdot (hU) = (gh) U}$

In Abelian groups , every subgroup is a normal subgroup. Thus, after each subgroup, the factor group can be formed there, which in turn is Abelian.

The order of the factor group is just the number of minor classes of . This number is called the index of in and is denoted by. If it is a finite group, then by Lagrange's theorem we have . ${\ displaystyle G / N}$${\ displaystyle N}$${\ displaystyle N}$${\ displaystyle G}$${\ displaystyle (G: N)}$${\ displaystyle G}$ ${\ displaystyle (G: N) = | G / N | = {\ tfrac {| G |} {| N |}}}$

## Examples

### Example ℤ 6

Let be the group of integers with addition as a group operation and be the subgroup of that consists of all multiples of 6. The group is Abelian and therefore every subgroup is a normal divisor . The factor group now consists of all secondary classes of the subgroup , these are: ${\ displaystyle \ mathbb {Z}}$${\ displaystyle 6 \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} / 6 \ mathbb {Z}}$${\ displaystyle 6 \ mathbb {Z}}$

${\ displaystyle 6 \ mathbb {Z} +0 = \ {\ dotsc, -18, -12, -6,0,6,12,18, \ dotsc \}}$

${\ displaystyle 6 \ mathbb {Z} +1 = \ {\ dotsc, -17, -11, -5,1,7,13,19, \ dotsc \}}$

${\ displaystyle 6 \ mathbb {Z} +2 = \ {\ dotsc, -16, -10, -4,2,8,14,20, \ dotsc \}}$

${\ displaystyle 6 \ mathbb {Z} +3 = \ {\ dotsc, -15, -9, -3,3,9,15,21, \ dotsc \}}$

${\ displaystyle 6 \ mathbb {Z} +4 = \ {\ dotsc, -14, -8, -2,4,10,16,22, \ dotsc \}}$

${\ displaystyle 6 \ mathbb {Z} +5 = \ {\ dotsc, -13, -7, -1,5,11,17,23, \ dotsc \}}$

These are all additional classes of how you can easily see, as they group partition and , , and so on. Since the operation is in addition, the addition of the secondary classes is also called addition and it applies, for example . One writes abbreviated ${\ displaystyle 6 \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle 6 \ mathbb {Z} + 6 = 6 \ mathbb {Z} +0}$${\ displaystyle 6 \ mathbb {Z} + 7 = 6 \ mathbb {Z} +1}$${\ displaystyle 6 \ mathbb {Z} + 8 = 6 \ mathbb {Z} +2}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle (6 \ mathbb {Z} +3) + (6 \ mathbb {Z} +4) = 6 \ mathbb {Z} + 7 = 6 \ mathbb {Z} +1}$

${\ displaystyle [0] = 6 \ mathbb {Z} +0}$,   ,   ,   ,   ,   ,${\ displaystyle [1] = 6 \ mathbb {Z} +1}$${\ displaystyle [2] = 6 \ mathbb {Z} +2}$${\ displaystyle [3] = 6 \ mathbb {Z} +3}$${\ displaystyle [4] = 6 \ mathbb {Z} +4}$${\ displaystyle [5] = 6 \ mathbb {Z} +5}$

thus consists of the 6 elements and results in the following table for the factor group${\ displaystyle \ mathbb {Z} / 6 \ mathbb {Z}}$${\ displaystyle [0], [1], [2], [3], [4], [5]}$${\ displaystyle \ mathbb {Z} _ {6}}$

${\ displaystyle +}$ ${\ displaystyle [0]}$ ${\ displaystyle [1]}$ ${\ displaystyle [2]}$ ${\ displaystyle [3]}$ ${\ displaystyle [4]}$ ${\ displaystyle [5]}$
${\ displaystyle [0]}$ ${\ displaystyle [0]}$ ${\ displaystyle [1]}$ ${\ displaystyle [2]}$ ${\ displaystyle [3]}$ ${\ displaystyle [4]}$ ${\ displaystyle [5]}$
${\ displaystyle [1]}$ ${\ displaystyle [1]}$ ${\ displaystyle [2]}$ ${\ displaystyle [3]}$ ${\ displaystyle [4]}$ ${\ displaystyle [5]}$ ${\ displaystyle [0]}$
${\ displaystyle [2]}$ ${\ displaystyle [2]}$ ${\ displaystyle [3]}$ ${\ displaystyle [4]}$ ${\ displaystyle [5]}$ ${\ displaystyle [0]}$ ${\ displaystyle [1]}$
${\ displaystyle [3]}$ ${\ displaystyle [3]}$ ${\ displaystyle [4]}$ ${\ displaystyle [5]}$ ${\ displaystyle [0]}$ ${\ displaystyle [1]}$ ${\ displaystyle [2]}$
${\ displaystyle [4]}$ ${\ displaystyle [4]}$ ${\ displaystyle [5]}$ ${\ displaystyle [0]}$ ${\ displaystyle [1]}$ ${\ displaystyle [2]}$ ${\ displaystyle [3]}$
${\ displaystyle [5]}$ ${\ displaystyle [5]}$ ${\ displaystyle [0]}$ ${\ displaystyle [1]}$ ${\ displaystyle [2]}$ ${\ displaystyle [3]}$ ${\ displaystyle [4]}$

This gives you a method with which you can construct subgroups like . ${\ displaystyle \ mathbb {Z} _ {6}}$

### Residual class group of the additive group of integers

The previous example can be generalized: For each there is a subgroup of the Abelian group , in particular a normal subgroup. The factor group is called the remainder class group modulo and is referred to for short with . It has exactly the elements. ${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle (n \ mathbb {Z}, +)}$${\ displaystyle (\ mathbb {Z}, +)}$${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$ ${\ displaystyle n}$${\ displaystyle \ mathbb {Z} _ {n}}$${\ displaystyle n}$

${\ displaystyle [k] _ {n}: = [k]: = k + n \ mathbb {Z} = \ {k + m \: \ m \ in n \ mathbb {Z} \} = \ {k + nz \: \ z \ in \ mathbb {Z} \}}$

and are called congruence classes with regard to addition modulo${\ displaystyle n}$ . So it is

${\ displaystyle \ mathbb {Z} / n \ mathbb {Z} = \ {[0], [1], \ ldots, [n-1] \}}$.

The inner connection of is usually referred to again with . In true, for example ${\ displaystyle \ mathbb {Z} / n \ mathbb {Z}}$${\ displaystyle +}$${\ displaystyle \ mathbb {Z} / 5 \ mathbb {Z}}$

${\ displaystyle \ [3] _ {5} + [4] _ {5} = [3] + [4] = [2] = [2] _ {5}}$,

there so . ${\ displaystyle 3 + 4 = 7 = 2 + 5}$${\ displaystyle (3 + 4) +5 \ mathbb {Z} = 2 + 5 \ mathbb {Z}}$

### Factor group according to nuclei of homomorphisms

Let and be two groups and a group homomorphism . Then the kernel of is a normal divisor of and therefore the factor group can be formed. According to the theorem of homomorphism for groups, this group of factors is isomorphic to the image of , which is a subgroup of . ${\ displaystyle G}$${\ displaystyle H}$${\ displaystyle \ varphi: G \ rightarrow H}$${\ displaystyle \ varphi}$${\ displaystyle G}$${\ displaystyle G / \ ker \ varphi}$${\ displaystyle \ varphi}$${\ displaystyle H}$

## Universal property of the factor group

Is a normal subgroup of , then the map with nuclear one epimorphism , so a surjective homomorphism . The universal property now states that for each group homomorphism with exactly one homomorphism with exists. ${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle \ pi \ colon G \ rightarrow G / H}$${\ displaystyle g \ mapsto gH}$${\ displaystyle H}$ ${\ displaystyle \ varphi \ colon G \ rightarrow G '}$${\ displaystyle H \ subseteq ker (\ varphi)}$${\ displaystyle \ varphi '\ colon G / H \ rightarrow G'}$${\ displaystyle \ varphi = \ varphi '\ circ \ pi}$

Example: Let be the natural projection of the whole numbers onto the residue class group modulo 6. Let be group homomorphism. Then lies at the core of and results in: ${\ displaystyle \ pi \ colon \ mathbb {Z} \ rightarrow \ mathbb {Z} / 6 \ mathbb {Z}}$${\ displaystyle \ varphi \ colon \ mathbb {Z} \ rightarrow \ mathbb {Z} / 3 \ mathbb {Z}}$${\ displaystyle 6 \ mathbb {Z}}$${\ displaystyle \ varphi}$${\ displaystyle \ varphi '\ colon \ mathbb {Z} / 6 \ mathbb {Z} \ rightarrow \ mathbb {Z} / 3 \ mathbb {Z}}$

${\ displaystyle \ varphi '([0] _ {6}) = [0] _ {3}}$

${\ displaystyle \ varphi '([1] _ {6}) = [1] _ {3}}$

${\ displaystyle \ varphi '([2] _ {6}) = [2] _ {3}}$

${\ displaystyle \ varphi '([3] _ {6}) = [0] _ {3}}$

${\ displaystyle \ varphi '([4] _ {6}) = [1] _ {3}}$

${\ displaystyle \ varphi '([5] _ {6}) = [2] _ {3}}$.

## Construction of groups

The transition to the factor group ensures that all elements of the normal divider are mapped to the neutral element. In this way one can force the existence of certain identities.

### Commutator group

Of all commutators group produced is a normal subgroup of the group . In the factor group , therefore, all commutators become trivial, i.e. the factor group is Abelian. This is called the abelization of the group. ${\ displaystyle [G, G]}$${\ displaystyle G}$${\ displaystyle G / [G, G]}$

### Relations

More generally, you can force any equations (relations) to exist in a group. If there are elements in the desired equations , consider the smallest normal divisor in the free group above elements , which contains all expressions in that are supposed to be equal to the neutral element. The factor group does what is required. More details can be found in the article " Presentation of a group ". ${\ displaystyle x_ {1}, \ ldots, x_ {n}}$ ${\ displaystyle F_ {n}}$${\ displaystyle n}$${\ displaystyle N}$${\ displaystyle x_ {1}, \ ldots, x_ {n}}$${\ displaystyle F_ {n} / N}$