 
Summary: Limits of Abelian Subgroups of Finite pgroups
J.L. Alperin \Lambda
University of Chicago
alperin@math.uchicago.edu
G. Glauberman y
University of Chicago
gg@math.uchicago.edu
1 Introduction
Abelian subgroups play a key role in the theory and applications of finite
pgroups. Our purpose is to establish some very general results motivated by
special results that have been of use. In particular, it is known [KJ] that if
a finite pgroup, for odd p, has an elementary abelian subgroup of order p n ,
n Ÿ 5, then it has a normal elementary abelian subgroup of the same order.
Our first main result is a general one of this type.
Theorem A
If A is an elementary abelian subgroup of order p n in a pgroup P , then
there is a normal elementary abelian subgroup B of the same order contained
in the normal closure of A in P , provided that p is odd and greater than
4n \Gamma 7.
We are very grateful to the referee for an improvement to our argument
