I am reading Dummit and Foote section 3.4 (Composition series and the Holder program) and I am having trouble understanding how these concepts are connected.
A group $G$ is simple if the only normal subgroups are $\{1\}$ and $G$, i.e., $G$ has no non-trivial normal subgroups.
A subnormal series of groups $$1=N_0 \lhd N_1 \lhd \cdots \lhd N_k=G$$ is called a composition series if $N_{i+1}/N_i$ is simple for all possible $i$.
From this I gather that if a group $G$ is simple then its composition series must be trivial, i.e., $\{1\} \le G$.
A group $G$ is solvable if there is a subnormal series $$1=G_0 \lhd G_1 \lhd \cdots \lhd G_s=G$$ such that $G_{i+1}/G_i$ is abelian for all possible $i$.
From this I gather that every abelian group is solvable. Also, if a simple group is solvable, then it must be abelian.
How are these concepts further related?