Suppose, $G$ is a finite group, $H \triangleleft G$, $G/H = K$. Suppose, $a=\min\{k |\forall A \subset H \text{ if }|A| > k\text{, then } AAA = \langle A \rangle \}$ Suppose, $b=\min\{k |\forall B \subset K \text{ if }|B| > k\text{, then } BBB = \langle B \rangle \}$ Suppose, $c=\min\{k |\forall C \subset G \text{ if }|C| > k\text{, then } CCC = \langle C \rangle \}$ Suppose, $|H|$, $|K|$, $a$ and $b$ are given. Is there a way to find a nontrivial upper bound for $c$ using them?
It is quite obvious, that $c \leq |G| = |H||K|$, but I need something stronger than that.
As the values of $a$, $b$ and $c$ are hard to calculate in general, the only example I know is: If $G = C_{pq}$, $H = C_p$ and $K = C_q$, where $p$ and $q$ are coprime, then $a = \lceil \frac{p}{3} \rceil + 1$, $b = \lceil \frac{q}{3} \rceil + 1$ and $c = \lceil \frac{pq}{3} \rceil + 1$. But that does not help much.
Any help will be appreciated.