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?