= Solution
A \b[<simple group>] is a nontrivial <group> whose only <normal subgroups> are itself and the trivial <subgroup>.
A \b[<composition series>] is a chain $G=G_0\triangleright G_1\triangleright\cdots\triangleright G_r=1$ in which each $G_{i+1}$ is a <normal subgroup> of $G_i$ and each <quotient group> $G_i/G_{i+1}$ is simple and nontrivial. Its length is $r$, and these quotients are its <composition factors>. The individual terms need not all be normal in $G$.
A \b[<solvable group>] is a <group> whose <derived series> reaches the trivial <group>: starting with $G^{(0)}=G$, define $G^{(i+1)}=[G^{(i)},G^{(i)}]$. For finite <groups> this is equivalent to having a <composition series> whose factors are all <cyclic groups> of <prime> order.
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