Quasi-Antichain Chermak-Delgado Lattices of Finite Groups

Abstract

The Chermak-Delgado lattice of a finite group is a dual, modular sublattice of the subgroup lattice of the group. This paper considers groups with a quasi-antichain interval in the Chermak-Delgado lattice, ultimately proving that if there is a quasi-antichain interval between $L$ and $H$ with $L \leq H$ then there exists a prime $p$ such that the quotient $H / L$ is an elementary abelian $p$-group and the number of atoms in the quasi-antichain is one more than a power of $p$. In the case where the Chermak-Delgado lattice of the entire group is a quasi-antichain, the relationship between the number of abelian atoms and the prime $p$ is examined; additionally several examples of group with a quasi-antichain Chermak-Delgado lattice are constructed.