On the lattice formed by all normal subloops of a finite loop (Q1900057)

It is known, that the lattice \(L(G)\) of all normal subgroups of a finite group \(G\) is distributive iff no two normal subgroups of \(G\) have equal orders. In the present paper, the author shows by an example that a similar statement is not true for loops. As a contradiction, the following theorem is proved: let \(G\) be a finite centrally nilpotent loop with Frattini subloop \(\Phi(G)\) and assume the normal subloop lattice \(L(G)\) is distributive. Then \(G\) is monogenic and \(G/\Phi(G)\) is a cyclic group. It is proved, that similar to the case of groups, 1) \(L(G)\) is a projective geometry iff \(G\) is an elementary abelian \(p\)-group for some prime \(p\); 2) \(L(G)\) is complemented iff \(G\) is a direct product of simple subloops. In particular, \(L(G)\) is a Boolean algebra iff \(G\) is a direct product of simple loops such that no two atoms of \(L(G)\) which are abelian groups are isomorphic.