A Determination of $B$-groups of Order $p^4$

Abstract

A group $H$ is said to be a $B$-group if whenever a primitive permutation group $G$ contains a regular subgroup isomorphic to $H$, then $G$ is doubly transitive. Going as far back as William Burnside, several authors have investigated whether certain families of groups are $B$-groups. Using the O'Nan-Scott Theorem as our starting point, we classify the $B$-groups of order $p^4$ where $p$ is a prime.