Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Embedding of abelian subgroups in $ p$-groups


Author: Marc W. Konvisser
Journal: Trans. Amer. Math. Soc. 153 (1971), 469-481
MSC: Primary 20.40
MathSciNet review: 0271228
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Research concerning the embedding of abelian subgroups in $ p$-groups generally has proceeded in two directions; either considering abelian subgroups of small index (cf. J. L. Alperin, Large abelian subrgoups of $ p$-groups, Trans. Amer. Math. Soc. 117 (1965), 10-20) or considering elementary abelian subgroups of small order (cf. B. Huppert, Endliche Gruppen. I, Springer-Verlag, Berlin, 1967, p. 303). The following new theorems extend these results:

Theorem A. Let $ G$ be a $ p$-group and $ M$ a normal subgroup of $ G$. (a) If $ M$ contains an abelian subgroup of index $ p$, then $ M$ contains an abelian subgroup of index $ p$ which is normal in $ G$. (b) If $ p \ne 2$ and $ M$ contains an abelian subgroup of index $ {p^2}$, then $ M$ contains an abelian subgroup of index $ {p^2}$ which is normal in $ G$.

Theorem B. Let $ G$ be a $ p$-group, $ p \ne 2, M$ a normal subgroup of $ G$, and let $ k$ be 2, 3, 4, or 5. If $ M$ contains an elementary abelian subgroup of order $ {p^k}$, then $ M$ contains an elementary abelian subgroup of order $ {p^k}$ which is normal in $ G$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20.40

Retrieve articles in all journals with MSC: 20.40


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0271228-5
PII: S 0002-9947(1971)0271228-5
Keywords: $ p$-group, abelian subgroup, elementary abelian subgroup, normal abelian subgroup, linear transformation, embedding of abelian subgroup
Article copyright: © Copyright 1971 American Mathematical Society