Different way to do Product Operation in a Symmetric Group in different texts

Question

In Herstein's Algebra, a product operation between any two elements(say $\sigma$ and $\tau$) of a symmetric group($S_3$) is defined as :

$\sigma = \begin{pmatrix} 1 & 2 & 3\\ 2 & 3 & 1 \end{pmatrix} $

$\tau= \begin{pmatrix} 1 & 2 & 3\\ 2 & 1 & 3 \end{pmatrix} $

$(1)\sigma\tau = (\sigma(1)\tau) = (2)\tau = 1$

Whereas, in other texts (Gross's lecture),

$\sigma\tau(1) = (\sigma\tau(1)) = \sigma(2) = 3$

I am guessing both are valid. But then are we talking about two different groups here depending on different ways of doing product operation. Or is it that the set $S_3$ is a unique group for both kind of product operation ?

So my question basically is, For a set, if it satisfies all groups condition under many binary operations, then are they different groups corresponding to different binary operation or the same group?

Answer 1

This is just whether you act on the left or on the right. You have composed $\sigma$ first, then $\tau$ on the first line, and on the second line you have done $\tau$ first, then $\sigma$. So of course you obtain a different answer. However, $$\tau\sigma(1)=\tau(\sigma(1))=\tau(2)=1.$$ So you obtain the same answer.