Question

In my book, i have:
"(xy)^n=xy*xy*xy....*xy=x^n y^n" And it says that the reason this is true is beacuse the group is abelian. What does a group being abeliean have to do with the way (xy)^n is multiplied out?

Answer 1

Let's look at a more specific case. Say
\[\Large (xy)^2\]

Answer 2

Literally, this means
\[\Large (xy)(xy) = xyxy \]right?

Answer 3

Okay

Answer 4

This will ONLY be equal to \[\Large x^2y^2 = x\color{red}{xy}y\]
If we were allowed to switch the positions of these two:
\[\Large x\color{blue}{yx}y\] in the first place.
IE, only if
\[\Large xy = yx \] to begin with, or the group was abelian.

Answer 5

Got it? :D

Answer 6

so, (xy)(xy)=xyxy, but that is only eequal to x^2y^2=xxyy, if we can change xyxy

Answer 7

if we can change the middle yx into an xy

Answer 8

okay, that makes sense, thank you