A generalisation of a theorem of Wielandt

In 1974, Helmut Wielandt proved that in a finite group G, a subgroup A is subnormal if and only if it is subnormal in 〈A,g〉 for all g∈G. In this paper, we prove that the subnormality of an odd order nilpotent subgroup A of G is already guaranteed by a seemingly weaker condition: A is subnormal in G if for every conjugacy class C of G there exists c∈C for which A is subnormal in 〈A,c〉. We also prove the following property of finite non-abelian simple groups: if A is a subgroup of odd prime order p in a finite almost simple group G, then there exists a cyclic p′-subgroup of F⁎(G) which does not normalise any non-trivial p-subgroup of G that is generated by conjugates of A.