Q:

Bounds on Size of Join of two Subgroups

Let $H$ and $K$ be two subgroups of a finite group $G$. Are there upper and lower bounds on the size of $H \vee K = \left<H \cup K\right>$ that depend on the size of the intersection $H \cap K$?
I believe that in case the intersection is trivial $(\left|H \cap K\right| = 1)$, then we have $\left|H \vee K\right| \geq \left|H\right|\left|K\right|$. Is this bound tight or even correct?
When the intersection is nontrivial there seem to be less elements in $H \vee K$, but I could not find a precise upper bound. Is there an upper bound in this case?

A:

Let $HK=\{hk\mid h\in H, k\in K\}$. Note that this is not necessarily a group, yet it is a basic exercise (found in most books) that $|HK|=\frac{|H||K|}{|H\cap K|}$.
Moreover, $HK\subseteq \langle H\cup K\rangle$, so $\frac{|H||K|}{|H\cap K|}\leq|\langle H\cup K\rangle|$.
In general, there is no meaningful upper bound on $|\langle H\cup K\rangle|$; $H$ and $K$ can both be very small but generate a large group.
On the other hand, if $HK$ is a group (which is always the case when one of $H$ or $K$ is normal in $G$, for example, or more generally, when one of them normalises the other) then $HK=\langle H\cup K\rangle$ and the previous inequality becomes an equality.

