Embedding of abelian subgroups in $p$-groups
HTML articles powered by AMS MathViewer
- by Marc W. Konvisser PDF
- Trans. Amer. Math. Soc. 153 (1971), 469-481 Request permission
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
- J. L. Alperin, Large Abelian subgroups of $p$-groups, Trans. Amer. Math. Soc. 117 (1965), 10–20. MR 170946, DOI 10.1090/S0002-9947-1965-0170946-4
- Walter Feit and John G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029. MR 166261
- Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 153 (1971), 469-481
- MSC: Primary 20.40
- DOI: https://doi.org/10.1090/S0002-9947-1971-0271228-5
- MathSciNet review: 0271228