gfaktor from sgnormal

8/6/2019 GFaktor From SGNormal

1/13


2/13

Introduction

Let Gb a group and let Hbe asubgroup of G.

Now Hhas both left cosets and

right cosets, and in general, a left

coset aHneed not be the sameas the right coset Ha.


3/13

Introduction (Cont.)

Suppose we try to define a binary operationon left cosets by defining

(aH)(bH) = (ab)H (1)

Eq. (1) is meaningless unless it gives a well-defined operation.

The theorem that follows shows that Eq. (1)gives a well-defined binary operation iff theleft and right cosets aHand Ha coincide forall a G.


4/13

Theorem

Let Hbe a subgroup of a group G.Then left cosets multiplication is

well-defined by the equation(aH)(bH) = (ab)H

if and only if left and right cosetscoincide, so that aH= Ha for alla G.


5/13

Proof

() Suppose first that (aH)(bH) = (ab)Hdoes give a well-defined binary operation

on left cosets.Let a G. We want to show that aH

and Ha are the same set.

We use the standard technique of showingthat each is a subset of the other.


6/13

Proof (cont.)

Letx aH. Choosing representativesx aH and a-1 a-1H, we have

(xH)(a-1H) = (xa-1)H.

On the other hand, choosing represen-tatives a aH and a-1 a-1H, we see that

(aH)(a-1H) = (aa-1)H= eH=H.

Using our asumption that left coset

multiplication by representatives is welldefined, we must have xa-1 = h H.Then x = ha so x Ha and aHHa.


7/13

Proof (cont.)

Now suppose that y Ha, so that y = h1a for

some h1 H. Then y-1 = a-1h1

-1 a-1H.

Since (a-1H)(aH) = Hand left multiplication is

assumed to be well-defined, we see that

y-1a = h2H so y = ah

2

-1.

This show that Ha aH.

Consequently aH = Ha.


8/13

Proof (cont.)

() If left and right cosets coincide, then leftcoset multiplication by representatives is welldefined.

Suppose we wish to compute (aH)(bH).

Choosing a aH and b bH, we obtain the

coset (ab)H.

Choosing different representatives ah1 aH

and bh2 bH we obtain the coset (ah1bh2)H.


9/13

Proof (cont.)

We must show that (ab)H and (ah1bh2)H arethe same coset.

Now h1b Hb = bH, so h1b = bh3 for someh3H. Thus

(ah1)(bh2) = a(h1b)h2 = a(bh3)h2= (ab)(h3h2)

and (ab)(h3h2) (ab)H .

Therefore (ah1bh2) (ab)H.


10/13

Proof of the cosets do form a group

with such coset multiplication is

identical to that given in the proof of

theorem about factor groups from

homomorphisms.


11/13

Corollary

Let H be a subgroup of a group G

whose left and right cosets

coincide. Then the cosets of H form

a group G/H under the binary

operation(aH)(bH) = (ab)H.


12/13

Definition

Factor Group The group G/H in

the preceding corollary is the

factor group (or quotient group) of

G modulo H.


13/13

Example

Since Z is an abelian group, nZ is a

normal subgroup. Then Z/nZ is a factor

group.

gfaktor from sgnormal

Documents