Does the subnormal subgroups always form a lattice?

Question

Let $G$ be a group and $W(G)$ the set of all subnormal subgroups of $G$, partially ordered by inclusion. My question is if $W(G)$ always forms a lattice (but not necessarily a sublattice of $L(G)$). I've heard it holds when $W(G)$ satisfies the maximal condition.

Answer 1

For finite groups $G$ the set $W(G)$ is a sublattice of $L(G)$, since the intersection $A\cap B$ of two subnormal subgroups is again subnormal in $G$, and the join $\langle A,B\rangle$ is again subnormal. For infinite groups, the first part remains true, but the last part need not be true. In fact, Zassenhaus gave an example where the join is not subnormal again. It was recognized that here chain conditions play an important role.