Jeffrey M. Riedl | Searcharxiv

arXiv2026

Linear $2$-Group Actions With Exactly One Regular Orbit

Let $G$ be a finite $2$-group and $V$ a finite faithful irreducible $G$-module. We show that if $G$ has exactly one regular orbit on $V$, then the characteristic $q$ of $V$ is a Fermat prime, $V$ is irreducible over ${\Bbb F}_q$, ${\rm dim}_{{\Bbb F}_q}V$ is a power of $2$, and the centraliser of $G$ in ${\rm GL}(V)$ embeds into $G$. For every Fermat prime $q$ and every $m\ge 1$ we determine the largest orbit size $2^{x_m}$ of a Sylow $2$-subgroup $P_m$ of ${\rm GL}_{2^{m-1}}(q)$ on its natural module $V_m$, and the number of orbits of this size, which is $1$, $2$, or $3$. Our main result is that for every Fermat prime $q$ and every $m\ge 1$ there is an irreducible subgroup of $P_m$ of order $2^{x_m}$ that has exactly one regular orbit on $V_m$. In this respect $p=2$ behaves like $p=3$, and unlike the Mersenne primes $p\ge 7$ studied in an earlier paper. The proof places a group related to a Singer cycle into every block of $q-1$ coordinates. This reduces the problem, for all Fermat primes at once, to a question about subgroups of a Sylow $2$-subgroup of a symmetric group acting on subsets, which we answer by means of fibre products. We also determine which largest orbits occur in two small levels, and we classify the examples in some small cases with the help of a computer.

math.GR↗

arXiv2026

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 ${\rm GL}_1(2^{n})\wr S_{p^{m-1}}$ and $V_m$ is its natural module, regarded over ${\Bbb F}_2$. 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.

math.GR↗