# Permutation multiplication in S3

I have a question regarding multiplication in S3 with a=(2 3) and b=(1 2)

The video I watched was showing left and right cosets in this case being different with

ab=(1 3 2)

ba=(1 2 3)

I understand ba as (1 2)(2 3)= (1↦2↦3)

but im not sure of ab as (2 3)(1 2)= 2↦3 but after this im stuck and have no idea. Im trying to think how ( 1 3 2) is the result of this but I have no idea

• $(12)(23)$ maps $1$ to $2$, $2$ to $3$, and $3$ to $2$ to $1$, so it's $(123)$ – J. W. Tanner 2 days ago

## 1 Answer

$$(23)(12)$$ maps $$1$$ to $$2$$ to $$3$$, $$3$$ to $$2$$, and $$2$$ to $$1$$, so it's $$(132)$$.