Let $A$ be a group with subgroup $H=P \times Q$, where $P$ is infinite and $Q$ is finite.
Suppose there exists $M \lhd_{f} A$ such that $p \notin M, q \notin M$, where $p \in P, q \in Q$.
Is it true that $pq \notin M$?
Try an example. Say for example $P=\mathbb{Z},Q=\mathbb{Z}_2$. Each uses addition as the group operator. M is the group of even integers (cross zero). $p=q=1$. Neither is in M. Their sum is in M.
If I change $P$ and $Q$, to $P=\langle p \rangle$, an infinite cyclic subgroup and $Q$ is finite central subgroup in $A$.
It is clear that $P \cap Q=1$.
can $pq \notin M$?
If I change $P$ and $Q$, to $P=\langle p \rangle$, an infinite cyclic subgroup and $Q$ is finite central subgroup in $A$.
It is clear that $P \cap Q=1$.
can $pq \notin M$?
I am a bit rusty with my algebra. I think I misunderstood the question. I will leave it to someone else to answer from here.