cosets剰余類じょうよるい and Lagrange's theorem

Residue classes of integers classify integers by remainders. In a group群ぐん, elements can also be classified using a subgroup部分群ぶぶんぐん as the standard. This classification is the group-theoretic notion of a coset剰余類じょうよるい.

1Left cosets剰余類じょうよるい

For a subgroup部分群ぶぶんぐん $$H$$H of a group $$G$$G and an element $$g\in G$$g\in G,

is called the left coset左剰余類ひだりじょうよるい of $$H$$H by $$g$$g.

The set obtained by multiplying from the right,

is called the right coset右剰余類みぎじょうよるい.

If the group is commutative, left cosets剰余類じょうよるい and right cosets剰余類じょうよるい coincide. In a general group, however, they need not coincide.

2Cosets partition a group

Any two left cosets剰余類じょうよるい are either equal or disjoint. Also, the collection of all left cosets剰余類じょうよるい covers all of $$G$$G.

This means that cosets剰余類じょうよるい divide $$G$$G into boxes of the same size.

3Lagrange's theorem

For a finite group $$G$$G and a subgroup部分群ぶぶんぐん $$H$$H,

holds. Here $$[G:H]$$[G:H] is the number of left cosets剰余類じょうよるい of $$H$$H.

Therefore,

This is Lagrange's theoremラグランジュの定理ていり.

4Concrete example

Let $$G=\mathbb{Z}/6\mathbb{Z}$$G=\mathbb Z/6\mathbb Z and $$H=\left\{\right[0],[3\left]\right\}$$H=\{[0],[3]\}. The order位数いすう of $$H$$H is 2. The cosets剰余類じょうよるい are

There are three such cosets剰余類じょうよるい, so $$\left|G\right|=6=3·2$$|G|=6=3\cdot2.

5What is preserved

When a group is divided into cosets剰余類じょうよるい, we no longer look at individual elements, but at boxes consisting of elements shifted by a subgroup部分群ぶぶんぐん. The size of the subgroup部分群ぶぶんぐん is preserved in every box. This shows that, in a finite group, the order位数いすう of a subgroup部分群ぶぶんぐん divides the order位数いすう of the group.

6Warning about quotient groups商群しょうぐん

The set of cosets剰余類じょうよるい can always be formed. However, it is not always possible to make that set into a group by multiplying cosets剰余類じょうよるい. To build a quotient group商群しょうぐん, the subgroup部分群ぶぶんぐん must be a normal subgroup正規部分群せいきぶぶんぐん.

7Proof supplement: coset剰余類じょうよるい partitions and Lagrange's theorem

Let $$H\text{[PARSE ERROR: Undefined("Command("le")")]}G$$H\le G. Two left cosets剰余類じょうよるい $$aH$$aH and $$bH$$bH are either disjoint or exactly equal.

First verify that the left cosets cover $$G$$G. For every $$g\in G$$g\in G, the identity $$e$$e lies in $$H$$H, so $$g=ge\in gH$$g=ge\in gH.

Proof. Suppose $$aH\cap bH\ne \text{[PARSE ERROR: Undefined("Command("varnothing")")]}$$aH\cap bH\ne\varnothing, and take $$x\in aH\cap bH$$x\in aH\cap bH. Then there exist $$h_1,h_2\in H$$h_1,h_2\in H such that $$x=a{h}_1=b{h}_2$$x=ah_1=bh_2. From this,

For any $$ah\in aH$$ah\in aH,

and since $$\left(b^{-1}a\right)h\in H$$(b^{-1}a)h\in H, we have $$ah\in bH$$ah\in bH. Thus $$aH\subseteq bH$$aH\subseteq bH. The same argument gives $$bH\subseteq aH$$bH\subseteq aH, so $$aH=bH$$aH=bH.

Also, the map

is a bijection. Surjectivity follows from the definition定義ていぎ of $$aH$$aH. Injectivity follows because if $$a{h}_1=a{h}_2$$ah_1=ah_2, then multiplying by $$a^{-1}$$a^{-1} from the left gives $$h_1=h_2$$h_1=h_2. This uses the existence of $$a^{-1}$$a^{-1}, that is, it uses that $$G$$G is a group.

Therefore, in a finite group $$G$$G, the group is partitioned into left cosets剰余類じょうよるい of equal size. If the number of left cosets剰余類じょうよるい is $$[G:H]$$[G:H], then

This is Lagrange's theoremラグランジュの定理ていり.

8Corollary: the order位数いすう of an element divides the order位数いすう of the group

Define the order位数いすう of an element $$g$$g to be the number $$|⟨g⟩|$$|\langle g\rangle| of elements in the cyclic subgroup it generates. For an element $$g$$g of a finite group $$G$$G, Lagrange's theorem gives

The number $$|⟨g⟩|$$|\langle g\rangle| is the order位数いすう of $$g$$g, so in a finite group, the order位数いすう of every element divides the order位数いすう of the group.

In particular, if $$\left|G\right|=p$$|G|=p is prime and $$g$$g is not the identity, then the order位数いすう of $$g$$g is not 1 and must divide $$p$$p, so it is $$p$$p. Hence $$G=⟨g⟩$$G=\langle g\rangle, and every group of prime order位数いすう is cyclic.

9Exercise link

10Summary

A group coset剰余類じょうよるい divides a group into equally sized parts using a subgroup. In a finite group, the orders of subgroups and individual elements divide the order of the group, and every group of prime order is cyclic. Constructing a quotient group additionally requires normality.