Let $G=\mathbb{Z}_{p^n}\times \ldots\times\mathbb{Z}_{p^n}$ be a direct sum of $m>1$ cyclic groups of order $p^n$. Suppose you have an automorphism $\alpha$ acting irreducibly on $\Omega_1(G)$, on $\Omega_2(G)/\Omega_1(G)$, etc... May we say that the order of $\alpha$ tends to infinity with $n\to\infty$?
Addendum
I forgot the condition $m>1$, since clearly this is not true for $m=1$.