Question

could someone please explain me how the left and the right cosets work its under the topic lagrange theorem it is so confusing in the book i dont understand it (N)

Answer 1

@satellite73

Answer 2

you have subgroup N of a group G
right coset is a set. for any \(g\in G\) put \(Ng=\{ng|n\in N\}\)

Answer 3

like in the book it says this

Example 1: Let G=S3 and H = { (1), (13)}. Then the left cosets of H in G are

(1)H = H,
(12)H = {(12), (12)(13)} = {(12), (132)} = (132)H,
(13)H = { (13), (1)} = H,
(23)H = { (23), (23)(13)} = {(23), (123)} = (123)H

like i dont know how that is happening

Answer 4

ok how are you writing S3? we can work this out step by step if you like

Answer 5

yeah thats the definition but i dont know how to apply it yeah could we please

Answer 6

uhh what do you mean how am i writing S3 thats all it says in the book:s

Answer 7

i mean what are you using for elements?
\(\{e,(12), (13), (23), (123), (132)\}\) ?

Answer 8

i assume that so it doesnt really say what elements we are using it just says G=S3

Answer 9

ok lets use those
then we have a subgroup H is { 1, (13)}

Answer 10

yess

Answer 11

lets pick something in G= S3, say lets pick \(g=(1,2) \)
and now compute \(gH=(12)H=\{(12)h|h\in H\}\)

Answer 12

okk

Answer 13

should be easy, because there are only two elements of H

Answer 14

yeah could you do the first one because i am lost in how to compute it then i will try the rest

Answer 15

so basically what the are doing is picking a element from the set S3?

Answer 16

(12)(1)=(12)
(12)(13)=(123)

so \((12)H=\{(12), (123)\}\)

Answer 17

that is all
different elements of G may give you different cosets

Answer 18

ok confused where did you get (12)(13) from ? werent we doing (12)H?

Answer 19

yes

Answer 20

you take you element of g and multiply it by every element of H

Answer 21

OOOO thats what we are doing lol

Answer 22

this example might be too simple because H has only 2 elements
if it had 8 elements you would have to do 8 multiplications

Answer 23

genius thanks a lot ok so just so i know i will try doing (132)H since its not in the book you could maybe tell me if i am doing it well

Answer 24

ok try it and see what you get
but note that (123) is an element in the coset (12)H so don't be too surprised at the answer

Answer 25

sorry i meant (132) is an element in ...

Answer 26

(132)H = { (132)(1) , (132)(12)} = { (132), (12)}

Answer 27

yes

Answer 28

ahh i see whats going on right here so woukld the right hand coset work the same way?

Answer 29

actually you wrote that in your question but i did not want to spoil it for you
you wrote

(12)H = {(12), (12)(13)} = {(12), (132)} = (132)H,

Answer 30

o lol i was wondering why it said that, but thanks a lot man

Answer 31

yw
later you will see learn that if the right coset is the same as the left coset, then the group is called "normal" and has special properties
that comes next

Answer 32

i mean "the subgroup is called "normal""

Answer 33

so for the right hand coset of H(12) would it be just {(1)(12), (13)(12)} = { (12), (123)} ?

Answer 34

yeah i have to review that for my midterm as well

Answer 35

and since (12)H is not = H(12) it is not normal right

Answer 36

exactly

Answer 37

alright sounds good thanks a lot for clearing it up

Answer 38

that is H is not "normal"
normal applies to subgroups, not cosets

Answer 39

yw

Answer 40

whats yw if you dont mind me asking lol

Answer 41

your welcome LOL nvm