Question

Question on groups and symmetries

Let G = {(x, y, z): x, y, z are elements on R } and define (x1, y1, z1)*(x2, y2, z2) = (x1+x2, y1+x1z2+y2, z1+z2)

1) Show that (G,*) is a group and that it is non-abelian

2) Show that (G,*) does not have any elements of finite order (other than the identity)

Answer 1

@satellite73 for this question it is telling us to show if its a group or not now we have to show if associativity, identitym and the inverse hold?

Answer 2

yeah have fun

Answer 3

alright you know how it says (G,*) i dont know what that * defines ?

Answer 4

you wrote what it defines

Answer 5

look carefully at the question you posted. it says right in it what "*" is

Answer 6

(x1, y1, z1)*(x2, y2, z2) = (x1+x2, y1+x1z2+y2, z1+z2) ?

Answer 7

that is what you wrote right?

Answer 8

yeahh

Answer 9

then that is what * is

Answer 10

associative is going to be totally straight forward but a real pain to write. you have to show
\[\left((x_1,y_1,z_1)*(x_2,y_2,z_2)\right)*(x_3,y_3,z_3)=(x_1,y_1,z_1)*\left((x_2,y_2,z_2)*(x_3,y_3,z_3)\right)\] it is going to be a lot of writing, but not much else

Answer 11

get a nice sharp pencil

Answer 12

lol and that is just (x1,y1,z1)*(x2+x3, y2+x2z3+y3, z2+z3)

Answer 13

and now we multiply this guy again to get (x1+x2+x3, y1+x1z2+x1z3+y2+x2z3+y3, z1+z2+z3)

Answer 14

is that all we do for associativity ?

Answer 15

@satellite73

Answer 16

you have to do it twice to show that they are the same

Answer 17

you did \((ab)c\) now you have to do \(a(bc)\) and show that they are equal
no doubt they will be

Answer 18

oooo ok ok yeah your right

Answer 19

oh now that i look closer it seems you did \(a(bc)\) so now you have to do \((ab)c\) i said it is a pain

Answer 20

loll

Answer 21

you just have to write it. there is nothing really else to do

Answer 22

yeah truee next to find the identity we this is what throws me off for identity ae=a and ea=a is our identity just zero?

Answer 23

sorry would it be 1 because anything multiplied to itself has to be that same specific number?

Answer 24

@Zarkon could you pleaseeeee help me on this questionn

Answer 25

i really need help on this final questionn

Answer 26

@Zarkon @TuringTest pleasee help

Answer 27

@satellite73 could you please look at this question and help me on this proof pleaseeee