- #1

- 3,134

- 8

Let H and K be subgroups of G. Show that [tex]|HK|=\frac{|H|\cdot |K|}{|H\cap K|}[/tex], whether or not HK is a subgroup of G.

Now, if [tex]H\cap K = \left\{1\right\}[/tex], where 1 is the identity element, then [tex]|H\cap K| = {1}[/tex], and, since [tex]HK=\left\{hk : h \in H \wedge k \in K\right\}[/tex], then [tex]|HK|=|H|\cdot|K|[/tex], and the fact is obvious. Assume [tex]H\cap K \neq \left\{1\right\}[/tex]. That's the place where I need a push. Thanks in advance.