Omega subgroups of powerful p-groups |
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Authors: | Gustavo A. Fernández-Alcober |
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Affiliation: | (1) Matematika Saila, Euskal Herriko Unibertsitatea, 48080 Bilbao, Spain |
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Abstract: | Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all i ≥ 0: (i) exp Ωi(G) ≤ p i for odd p, and expΩi(G) ≤ 2 i+1 for p = 2; (ii) the index |G: G p i| coincides with the number of elements of G of order at most p i. Supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the University of the Basque Country, grant UPV05/99. |
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