Exercise 17, § 4

Let G be a group with operators; a composition series $(G_i)$ of G is called normal if all the $G_i$ are normal stable subgroups of G; a normal series $\Sigma$ is called principal if it is strictly decreasing and there exists no normal series distinct from $\Sigma$, finer than $\Sigma$ and strictly decreasing.

(a) If $(G_i)$ and $(H_j)$ are two normal series of G, show that there exist two equivalent normal series finer respectively than $(G_i)$ and $(H_j)$ (apply Schreier's Theorem by considering a suitable domain of operators for G). Give a second proof of this proposition by "inserting" the subgroups $G'_{ij} = G_i \cap (G_{i+1}H_j)$ and $H'_{ji} = H_j \cap (H_{j+1}G_i)$ in the sequences $(G_i)$ and $(H_j)$ respectively.

(b) If G has a principal series, any two principal series of G are equivalent; for every strictly decreasing normal series $\Sigma$ there exists a principal series finer than $\Sigma$. Deduce that, for G to possess a principal series, it is necessary and sufficient that the set of normal stable subgroups of G satisfy the maximal and minimal conditions.

(c) If G is a group without operators and possesses a principal series and if the set of subgroups of G satisfies the minimal condition, every quotient group

G_i/G_{i+1} is the direct product of a finite number of simple subgroups isomorphic to one another (use Exercise 15).