Am trying to answer the following exercise in "A First Graduate Course in Abstract Algebra" by W. J. Wickless, page 25, Fifth Problem Set, Problem 1.7.3 :
"Prove that any normal series for a group $G$ can be refined, by adding terms, to produce a composition series for $G$."
Wickless defines a composition series as a finite normal series in which every successive factor group is a simple group.
This does not seem to hold for infinite groups.
Consider the additive cyclic group $\bf{Z}$ of integers, and let $p$ be a prime number. In this group, the series $${0}\triangleleft{p{\bf{Z}}}\triangleleft{\bf{Z}}$$ is a normal series, but I cannot refine this to a composition series with a finite number of terms because the series $$\cdots\triangleleft{p^n{\bf{Z}}}\triangleleft{p^{n-1}{\bf{Z}}}\triangleleft\cdots\triangleleft{p{\bf{Z}}}\triangleleft{\bf{Z}}$$ as $n$ runs through the positive integers has every factor group being cyclic of order $p$ and therefore being simple. If we replace the prime $p$ by a prime $q\neq{p}$, then an identical series with $q$ in place of $p$ can be formed and these are not equivalent.
Do I need to assume that given group $G$ is finite? Am I missing something in these definitions?
Thanks!