Saturated fusion systems are categories modeling properties of conjugacy of p-elements in finite groups. It was shown by Chermak that there are group-like structures called regular localities associated to saturated fusion systems. Both the theory of fusion systems and the theory of regular localities are developed in analogy to the theory of finite groups. In this paper we focus on a classical theorem of Wielandt, which states that any two subnormal subgroups of a finite group G generate a subnormal subgroup of G. We prove versions of this theorem for regular localities and for fusion systems. Along the way we prove also a purely group-theoretical result which may be of independent interest.

A subgroup R of a finite group G is called weakly subnormal in G if R is not subnormal in G but it is subnormal in every proper overgroup of R in G. In this paper, weak subnormality is used to construct a subgroup lattice of a finite soluble group containing the lattice of all subnormal subgroups. A new characterisation of Schmidt groups is also obtained: they are exactly those groups with all subgroups subnormal or weakly subnormal.

We gather evidence on a new local-global conjecture of Moretó and Rizo on values of irreducible characters of finite groups. For this we study subnormalisers and picky elements in finite groups of Lie type and determine them in many cases, for unipotent elements as well as for semisimple elements of prime power order. We also discuss subnormalisers of unipotent and semisimple elements in connected as well as in disconnected reductive linear algebraic groups.

For a formation $\mathfrak F$, a subgroup M of a finite group G is said to be $\mathfrak F$-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉$\mathfrak F$ such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$-normal in G if G/CoreG(M) belongs to $\mathfrak F$. A subgroup U of a finite group G is called K-$\mathfrak F$-subnormal in G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or $\mathfrak F$-normal in Ui, for i = 1,2, …, n. We call a finite group G an $fT_{\mathfrak F}$-group if every K-$\mathfrak F$-subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations $\mathfrak F$ the structure of $fT_{\mathfrak F}$-groups. We pay special attention to the $\mathfrak F$-pronormal subgroups in this analysis.

Characterisations of finite groups in which normality is a transitive relation are presented in the paper. We also characterise the finite groups in which every subgroup is either permutable or coincides with its permutiser as the groups in which every subgroup is permutable.