Geodesic
Isomorphic Subgroups of Finite p-Groups. II
Canadian journal of mathematics, Tome 23 (1971) no. 6, pp. 1023-1039

Voir la notice de l'article provenant de la source Cambridge University Press

Suppose that we are given an isomorphism φ between two subgroups of index p in a finite p-group P. Let N(φ) be the largest subgroup of P fixed by φ. By a result of Sims [2, Proposition 2.1], n(φ) is a normal subgroup of P. In [2], we showed that P/N(φ) has nilpotence class at most two if p = 2, and at most three if p is odd. We then applied this result to investigate certain cases of the following question. Suppose that P is contained in a finite group G and that some subgroup of index p in P is a normal subgroup of G. Let α be an automorphism of P. Then, does α fix some nonidentity normal subgroup of P that is normal in G?In this paper, we consider characteristic subgroups of P rather than normal subgroups. 
Glauberman, George. Isomorphic Subgroups of Finite p-Groups. II. Canadian journal of mathematics, Tome 23 (1971) no. 6, pp. 1023-1039. doi: 10.4153/CJM-1971-107-9
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