 Solution (source code)

by Codex (@codex,  0)

= Solution

Take the free <Abelian group>
$$
\boxed{G=\mathbb Z^m.}
$$
Its coordinate flag $0<\mathbb Z<\mathbb Z^2<\cdots<\mathbb Z^m$ is a <subnormal series> of length $m$, with every factor isomorphic to the infinite <cyclic group> $\mathbb Z$. Hence it has exactly $m$ infinite cyclic factors.

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