Group of order 81 acting faithfully on a set of order 9

Question

I have to prove that if $G$ is a group of order 81 acting faithfully on a set $S$ of order 9, then $G$ acts transitively on $S$.

I have tried with the cardinality of the orbits. For example, if there's some orbit with order 9 the problem is finished. If not, the others possibilities are cardinalities 3 and 1. Then, I'd like to get some contradiction with the class equation, studying the partitions of the integer 9, but I cannot.

Can anyone help me with the solution? Maybe a hint for to solve it by myself.

Answer 1

$\newcommand{\Size}[1]{\left\lvert #1 \right\rvert}$ Hint 1

Suppose $G$ does not act transitively. Then the orbits have lengths $1$ or $3$.

Hint 2

If an orbit has length $3$, then the stabilizer has index $3$, and thus it is normal, so that it is the same for the three elements of the orbit.

Hint 3

Note the little lemma. If $H, K$ are subgroups of the finite group $G$, then $\Size{G : H \cap K} \le \Size{G:H} \Size{G:K}$.

Hint 4

If $G$ has three orbits of length $3$, then it is not faithful - this is because the intersection of the three stabilizers, one for each orbit, will have index at most $3^{3} = 27$.

Hint 5

The situation is even worse if there is an orbit (and thus at least three) of length $1$, and thus stabilizer $G$.