I need a definition of a transitive group that's accessible to someone who's just started learning group theory (so won't know about actions and orbits etc.). I've written the following:
A permutation group $S_X$ is transitive if there exists an element $\alpha \in S_X$ such that $x^{\alpha}=y$ for all $x, y \in X$.
Is there anything wrong with this? Furthermore, is the notation I've used what you'd usually see or could it be improved to be something more conventional? Thanks in advance!
A group $G$ of permutations on a set $X$ is transitive if for every pair of elements $x, y$ of $X$, there is a $\sigma$ in $G$ such that $\sigma(x) = y$ (or $x^\sigma = y$, if you prefer that notation). (I've avoided using your notation $S_X$, which strongly suggests the group of all permutations of $X$, which always acts transitively.) So, for example, the subgroup of $S_4$ generated by the transpositions $(1\,2)$ and $(3\,4)$ is not transitive, because it contains no permutation mapping $1$ to $3$.