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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.
Let $q$ be a power of a prime $p$, let $G$ be a finite Chevalley group over $mathbb{F}_q$ and let $U$ be a Sylow $p$-subgroup of $G$; we assume that $p$ is not a very bad prime for $G$. We explain a procedure of reduction of irreducible complex chara
We prove that a finite group $G$ has a normal Sylow $p$-subgroup $P$ if, and only if, every irreducible character of $G$ appearing in the permutation character $({bf 1}_P)^G$ with multiplicity coprime to $p$ has degree coprime to $p$. This confirms a
For $G={rm GL}(n,q)$, the proportion $P_{n,q}$ of pairs $(chi,g)$ in ${rm Irr}(G)times G$ with $chi(g) eq 0$ satisfies $P_{n,q}to 0$ as $ntoinfty$.
Denote by $ u_p(G)$ the number of Sylow $p$-subgroups of $G$. It is not difficult to see that $ u_p(H)leq u_p(G)$ for $Hleq G$, however $ u_p(H)$ does not divide $ u_p(G)$ in general. In this paper we reduce the question whether $ u_p(H)$ divides $ u
We determine the multiplicities of irreducible summands in the symmetric and the exterior squares of hook representations of symmetric groups over an algebraically closed field of characteristic zero.