Normal \(p\)-subgroups of the automorphism group of an Abelian \(p\)-group (Q1379067)

Let \(p\) be a prime and let \(G\) be an abelian \(p\)-group. Let \(\Pi=O_p(\Aut G)\) denote the maximal normal \(p\)-subgroup of the automorphism group \(\Aut(G)\) of \(G\). For the case that \(p\geq 5\), a description of \(\Pi\) in terms of its action on \(G\) was given by the reviewer [J. Lond. Math. Soc., II. Ser. 5, 409-413 (1972; Zbl 0246.20047)]\ and later used to demonstrate how to recover the finite Ulm invariants of \(G\) given \(\Aut G\) as an abstract group [J. Algebra 44, 9-28 (1977; Zbl 0345.20054)]. Recently, \textit{J. Hausen} and \textit{Ph. Schultz} have extended the description of \(\Pi\) to all primes [Proc. Am. Math. Soc. (to appear)]. In the current paper, the author examines the layer structure of \(\Pi\): for the case that \(p\neq 2\), he shows that, given any integer \(n\geq 0\), the automorphism group of \(G\) contains a unique \(p\)-subgroup which is maximal with respect to being normal and having exponent at most \(p^n\). Examples are provided to show that this need not be the case when \(p=2\). For \(p>2\), the author characterizes these maximal normal \(p^n\)-bounded \(p\)-subgroups of \(\Aut G\) in terms of their action on \(G\) and uses his results to show how the finite Ulm invariants of \(G\) may be recaptured from \(\Aut G\).