Overview
This research investigates the $p$-parts of stabilizers within primitive permutation groups. Specifically, it examines the finite set $\Omega$ acted upon by a primitive permutation group $G$. The study focuses on cases where the square of a prime $p$, denoted as $p^2$, divides the order of the group $|G|$.
Research Context
The study operates within the mathematical domain of permutation groups, a fundamental area of abstract algebra. Primitive permutation groups are a class of permutation groups with specific properties concerning their action on a set. Stabilizers are subgroups consisting of elements that fix certain points or subsets under the group's action. The $p$-part of a group's order, often denoted as $|G|_p$, refers to the highest power of the prime $p$ that divides the order of the group.
Approach
The research first addresses the case where the primitive permutation group $G$ is solvable. For this scenario, it aims to establish the existence of a specific type of subset within $\Omega$. Following this, the study indicates a different strategy for groups that are not solvable.
Findings
- When $G$ is a solvable primitive permutation group on a finite set $\Omega$, and $p^2$ divides $|G|$ for a prime $p$, there exists a subset of $\Omega$.
- The stabilizer $S$ of this subset has the property that $1 < |S|_p < |G|_p$. This implies that the $p$-part of the stabilizer's order is strictly between 1 and the $p$-part of the full group's order.
- For cases where $G$ is not solvable, a counting argument is proposed as a potentially helpful approach.