Let's say I have finite group $G$. Let it have have following subgroups:
$<a> = \{e, a\}$, $<b> = \{e, b\}$, $<c> = \{e, c, c^2\}$, $<d> = \{e, d, d^2, d^3\}$
I can for sure say that $ad, ad^2, ad^3$ are distinct.
Can I say that $bc$ is distinct too?
Or can I say the same about $bd, bd^2, bd^3$?
What is the least possible order of $G$ in this case?
also
I want to make this as general as possible, so how can we get new distinct elements if we know some of them?