Embedding of abelian subgroups in $p$-groups

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 $G$.