One subgroup divisible by another

Question

What does it mean for one subgroup $A \subset G$ to be divisible by another $B \subset G$?

I think that would also clarify the gcd and lcm of two subgroups for me.

This is a translation from one of the Sitzungsberichte by Frobenius, I don't know if this is modern terminology.

Answer 1

Frobenius introduces in Über endliche Gruppen, 1895 the following terminology for two sets that a part of a larger group:

• A set $B$ is divisible by $A$ iff $A \subset B$.
• The lcm of $A$ and $B$ is the smallest group containing $A \cup B$ (or $A B$).
• The gcd of $A$ and $B$ is therefore (the group) $A \cap B$.

(He then shows in Thm I, §.I, that $|A B| = |A| |B| / |A \cap B|$.)