Exercise 26, § 6

Let G be a finite group. We say that G is hyper-solvable if there exists a composition sequence $(G_i)_{0 \leq i \leq n}$ of G formed of normal subgroups of G such that the quotients $G_i/G_{i+1}$ are cyclic. (a) Now prove that every subgroup, every quotient group and every finite product of hyper-solvable groups is hyper-solvable. (b) Now prove that nilpotent $\Rightarrow$ hyper-solvable $\Rightarrow$ solvable, and that the converse implications are false. (c) Suppose that G is hyper-solvable. Now prove that, if $G \neq \{1\}$, there exists a normal subgroup C of G which is cyclic of prime order. Now prove that the derived D(G) of G is nilpotent. Now prove that every subgroup of G distinct from G, and maximal, has prime index. (d) We say that a finite group G is bicyclic if there exist two elements a, b of G such that, if one denotes by $C_a$ and $C_b$ the cyclic subgroups generated by a and b respectively, one has $G = C_a C_b = C_b C_a$. Now prove that every bicyclic group is hyper-solvable. (Reduce to proving that there exists in G a normal cyclic subgroup $N \neq \{1\}$. One may restrict oneself to the case where $C_a \cap C_b = \{1\}$; let $m \geq n > 1$ be the orders of a and b respectively; by considering the elements $b^{-1} a^k$, where $0 \leq k \leq m-1$, show that there exists an integer s such that $a^s \neq 1$ and $b^{-1} a^s b \in C_a$; the set of the $a^s$ having this property is the cyclic subgroup N sought.)