Let G be a finite group with Sylow 2-subgroup P 6 G. Navarro–Tiep–Vallejo have conjectured that the principal 2-block of NG(P) contains exactly one irreducible Brauer character if and only if all odd-degree ordinary irreducible characters in the principal 2-block of G are fixed by a certain Galois automorphism σ. By recent work of Navarro–Vallejo it suffices to show this conjecture holds for al...

a subgroup $h$ is said to be $s$-permutable in a group $g$‎, ‎if‎ ‎$hp=ph$ holds for every sylow subgroup $p$ of $g$‎. ‎if there exists a‎ ‎subgroup $b$ of $g$ such that $hb=g$ and $h$ permutes with every‎ ‎sylow subgroup of $b$‎, ‎then $h$ is said to be $ss$-quasinormal in‎ ‎$g$‎. ‎in this paper‎, ‎we say that $h$ is a weakly $ss$-quasinormal‎ ‎subgroup of $g$‎, ‎if there is a normal subgroup ...