Isomorphic Subgroups of Finite p-Groups. II
1971 ◽
Vol 23
(6)
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pp. 1023-1039
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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.
2013 ◽
Vol 12
(05)
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pp. 1250204
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1984 ◽
Vol 27
(1)
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pp. 7-9
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1985 ◽
Vol 37
(5)
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pp. 934-962
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1953 ◽
Vol 5
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pp. 477-497
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1969 ◽
Vol 21
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pp. 418-429
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2014 ◽
Vol 57
(3)
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pp. 648-657
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2016 ◽
Vol 16
(08)
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pp. 1750160