Geodesic
Isomorphic Subgroups of Finite p-Groups. I
Canadian journal of mathematics, Tome 23 (1971) no. 6, pp. 983-1022

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Let p be a prime and P be a p-subgroup of a finite group G. Suppose that g ∈ G and that P ∩ P g has index p in P. In [4], we assumed that g normalizes no non-identity normal subgroup of P. We obtained some bounds on the order of P and some applications to the case in which p = 2 and P is a Sylow 2-subgroup of 〈P, P g 〉. In this paper, we examine this situation further by considering the isomorphism φ of P ∩ P g-l onto P ∩ Pg given by φ(x) = xg . We actually consider arbitrary isomorphisms φ between two subgroups of index p in P. However, an easy argument (Lemma 2.3) shows that every such φ can be obtained as above for some G and some g. We obtain some results concerning the nilpotence class rather than the order of P. 
Glauberman, George. Isomorphic Subgroups of Finite p-Groups. I. Canadian journal of mathematics, Tome 23 (1971) no. 6, pp. 983-1022. doi: 10.4153/CJM-1971-106-0
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