I was taught the following group theory theorem:
Let H and K be subgroups of a group G and assume HK = KH $H \vee K=HK=KH$
which sometimes is called the product theorem (Ledermann, Introduction to group theory).
Afterwards, I studied the following theorem:
Let H and K be subgroups of a group G then $H \vee K=\{h_1k_1h_2k_2...h_rk_r | h_i \in H,k_i \in K, r \ge 1\}$
which seems to be a generalization of the first one. To my intuition it says that removing commutative properties is substituted by infinite repetition.
¿Do you know if this result has a name?¿Do you know any applications of it?