Linear Group Actions with Exactly One Regular Orbit

Let $p$ be an odd prime, $G$ a finite $p$-group, and $V$ a finite faithful irreducible $G$-module. We show that if $G$ has exactly one regular orbit on $V$, then $p=2^n-1$ is a Mersenne prime and $(G,V)$ embeds into $(P_m,V_m)$ for some $m\ge 1$, where $P_m$ is a Sylow $p$-subgroup of the monomial group ${\mathrm{G}\mathrm{L}}_1(2^n)\wr S_{p^{m-1}}$ and $V_m$ is its natural module, regarded over ${\mathbb{F}}_2$. We determine the largest orbit size $p^{x_m}$ of $P_m$ on $V_m$ and the number of orbits of this size, and we construct subgroups of $P_m$ of order $p^{x_m}$ whose unique regular orbit is a largest orbit of $P_m$; for $p=3$ such subgroups exist for every $m$, whereas for Mersenne primes $p\ge 7$ we prove that they exist only for $m\ge 3$. Finally, we classify the subgroups of $P_m$ of order $p^{x_m}$ with exactly one regular orbit for $m=2$ and, with the help of a computer, for $p=3$ and $m=3$, where there are $13$ such pairs up to isomorphism; for $p=3$ and $m=4$ we show that there are exactly $4914$ conjugacy classes of such subgroups in $P_4$. In particular, such a subgroup is in general not determined by its regular orbit.