A note on the structure of a finite group G having a subgroup H maximal in 〈H, Hg〉

Let G be a finite group and H ≤ G. The authors study the structure of finite groups G having a subgroup H which is maximal in 〈H, Hg〉 for some g ∈ G. Some results on the structure of 〈H, Hg〉 and G are set up. Especially, a generalization of Baer′s theorem is established.