Research concerning the embedding of abelian subgroups in p p -groups generally has proceeded in two directions; either considering abelian subgroups of small index (cf. J. L. Alperin, Large abelian subrgoups of p 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 G be a p p -group and M M a normal subgroup of G G . (a) If M M contains an abelian subgroup of index p p , then M M contains an abelian subgroup of index p p which is normal in G G . (b) If p ≠ 2 p \ne 2 and M M contains an abelian subgroup of index p 2 {p^2} , then M M contains an abelian subgroup of index p 2 {p^2} which is normal in G G . Theorem B. Let G G be a p p -group, p ≠ 2 , M p \ne 2, M a normal subgroup of G G , and let k k be 2, 3, 4, or 5. If M M contains an elementary abelian subgroup of order p k {p^k} , then M M contains an elementary abelian subgroup of order p k {p^k} which is normal in G G .