Linear characters of Sylow subgroups of symmetric groups

TL;DR

This paper characterizes when induced linear characters from Sylow p-subgroups of symmetric groups are equal, showing they are equal if and only if the original characters are conjugate under the normalizer, extending Navarro's result.

Contribution

It provides a new criterion for the equality of induced linear characters in symmetric groups, generalizing Navarro's theorem for p-solvable groups.

Findings

Induced characters are equal iff original characters are N-conjugate.

Extends Navarro's result to symmetric groups.

Provides a characterization of linear characters of Sylow subgroups.

Abstract

Let $p$ be any prime. Let $P_n$ be a Sylow $p$-subgroup of the symmetric group $S_n$. Let $\phi$ and $\psi$ be linear characters of $P_n$ and let $N$ be the normaliser of $P_n$ in $S_n$. In this article we show that the inductions of $\phi$ and $\psi$ to $S_n$ are equal if, and only if, $\phi$ and $\psi$ are $N$--conjugate. This is an analogue for symmetric groups of a result of Navarro for $p$-solvable groups.