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 subgroups L and H with L ≤ H then there exists a prime p such that 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 groups with a quasi-antichain Chermak–Delgado lattice are constructed.
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Dedicated to Otto H. Kegel for the occasion of his eightieth birthday.
The research of the second author was supported by Projecto MTM2010-19938-C03-02, Ministerio de Ciencia e Innovación, Spain. The research of the third author was supported by a Scholarly and Creative Activity Grant from Oswego State University.
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Brewster, B., Hauck, P. & Wilcox, E. Quasi-antichain Chermak–Delgado lattices of finite groups. Arch. Math. 103, 301–311 (2014). https://doi.org/10.1007/s00013-014-0696-3
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DOI: https://doi.org/10.1007/s00013-014-0696-3