11.104 (1990)

Let $G$ be a finite group of order $p^a \cdot q^b \dots$, where $p, q, \dots$ are distinct primes. Introduce distinct variables $x_p, x_q, \dots$ corresponding to $p, q, \dots$ . Define functions $f, \phi$ from the lattice of subgroups of $G$ to the polynomial ring $\mathbb{Z}[x_p, x_q, \dots]$ as follows: (1) if $H$ has order $p^\alpha \cdot q^\beta \dots$, then $f(H) = x_p^\alpha \cdot x_q^\beta \dots$; (2) for all $H \leqslant G$, we have $\sum_{K \leqslant H} \phi(K) = f(H)$. Then $f(G), \phi(G)$ may be called the order and Eulerian polynomials of $G$. Substituting $p^m, q^m, \dots$ for $x_p, x_q, \dots$ in these polynomials we get the $m$th power of the order of $G$ and the number of ordered $m$-tuples of elements that generate $G$ respectively. It is known that if $G$ is $p$-solvable, then $\phi(G)$ is a product of a polynomial in $x_p$ and a polynomial in the remaining variables. Consequently, if $G$ is solvable, $\phi(G)$ is the product of a polynomial in $x_p$ by a polynomial in $x_q$ by . . . . Are the converses of these statements true
$\qquad$ a) for solvable groups?
$\qquad$ b) for $p$-solvable groups?