I'm studying transitive and regular groups comparing the theorems and corollaries on several texts I have. One reference for sure, is Dixon's Permutation Groups. Actually, I stumbled into a corollary about transitive groups at p.8, sounding like this:
Suppose that $G$ is transitive in its action on the set $\Omega$. Then:
(i) the stabilizers $G_\alpha (\alpha\in\Omega)$ form a single conjugacy class of subgroups of $G$
(ii) The index $|G : G_\alpha| = |\Omega|$ for each $\alpha$
(iii) If $G$ is finite then the action of $G$ is regular $\iff |G|=|\Omega|$
While I ha no problems with (i) and (ii), that $\iff$ is disturbing me a bit. As far as I can it means "if and only if" made by $a\Rightarrow b$ and $a\Leftarrow b$. In our case this would read:
regularity $\Rightarrow |G|=|\Omega|$ and $|G=\Omega| \Rightarrow$ regularity
On other sources, I found a less strong implication, i.e. only: regularity $\Rightarrow |G|=|\Omega|$
I know a regular group shows $G_\alpha = {e}, \forall a\in\Omega$ so the order matching should be related to the number of cosets of the trivial stabilizers, but I don't exactly how to prove this "feeling".
May you help me on that, please? Thanks
For definitions, it is traditional to use "if" to mean "iff". See
Are "if" and "iff" interchangeable in definitions?