Find $l_{\mathbb{Z}}((\mathbb{Z} / \mathbb{Z}20)\oplus (\mathbb{Z} / \mathbb{Z}27))$.
Determine a composition series for the $\mathbb{Z}$-module $(\mathbb{Z}/\mathbb{Z}20)\oplus (\mathbb{Z} / \mathbb{Z}27)$
It's easy to see that $(\mathbb{Z} / \mathbb{Z}20)$ has a composition series $$0=(\mathbb{Z}20 / \mathbb{Z}20) \subset (\mathbb{Z}10 / \mathbb{Z}20) \subset (\mathbb{Z}5 / \mathbb{Z}20) \subset (\mathbb{Z} / \mathbb{Z}20)$$ Similarly $(\mathbb{Z}27 / \mathbb{Z}27) \subset (\mathbb{Z}9 / \mathbb{Z}27) \subset (\mathbb{Z}3 / \mathbb{Z}27) \subset (\mathbb{Z} / \mathbb{Z}27)$
But I don't know how to find a composition of $(\mathbb{Z}/\mathbb{Z}20)\oplus (\mathbb{Z} / \mathbb{Z}27)$
Let $A$ and $B$ be finite length modules, that is, admitting composition series. If $0=A_0\subset A_1\subset\dots\subset A_m=A$ and $0=B_0\subset B_1\subset\dots\subset B_n=B$ are composition series, then $$ 0=A_0\oplus0\subset A_1\oplus0\subset\dots\subset A_n\oplus 0 \subset A\oplus B_1\subset\dots\subset A\oplus B_n $$ is a composition series for $A\oplus B$.