Exercise 26, § 6

Let $G$ be a finite group. $G$ is called supersolvable if there exists a composition series $(G_i)_{0 \leq i \leq n}$ of $G$ consisting of normal subgroups of $G$ such that the quotients $G_i/G_{i+1}$ are cyclic. (a) Show that every subgroup, every quotient group and every finite product of supersolvable groups is supersolvable. (b) Show that nilpotent $\Rightarrow$ supersolvable $\Rightarrow$ solvable and that the converse implications are false.

(c) Suppose G is supersolvable. Show that, if $G \neq \{1\}$, there exists a normal subgroup C of G which is cyclic of prime order. Show that the derived group D(G) of G is nilpotent. Show that every subgroup of G distinct from G which is maximal is of prime index.

(d) A finite group G is bicyclic if there exist two elements $a, b$ of G such that, if $C_a$ and $C_b$ denote the cyclic subgroups generated by $a$ and $b$ respectively, then $G = C_a C_b = C_b C_a$. Show that every bicyclic group is supersolvable. (Reduce it to proving that there exists in G a normal cyclic subgroup $N \neq \{1\}$. Attention may be confined 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 $a^s$ with this property is the desired cyclic subgroup N.)