Corpus ID: 235755477

Groups whose Chermak-Delgado lattice is a subgroup lattice of an elementary abelian $p$-group

@inproceedings{An2021GroupsWC,
  title={Groups whose Chermak-Delgado lattice is a subgroup lattice of an elementary abelian \$p\$-group},
  author={Lijian An},
  year={2021}
}
The Chermak-Delgado lattice of a finite group G is a self-dual sublattice of the subgroup lattice of G. In this paper, we focus on finite groups whose ChermakDelgado lattice is a subgroup lattice of an elementary abelian p-group. We prove that such groups are nilpotent of class 2. We also prove that, for any elementary abelian p-group E, there exists a finite group G such that the Chermak-Delgado lattice of G is a subgroup lattice of E. 

References

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ABSTRACT It is an open question in the study of Chermak-Delgado lattices precisely which finite groups G have the property that ğ’žğ’Ÿ(G) is a chain of length 0. In this note, we determine two classes… Expand
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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… Expand
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For a finite group G with subgroup H the Chermak-Delgado measure of H in G refer to the product of the order of H with the order of its centralizer, C_G(H). The set of all subgroups with maximal… Expand
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Abstract The Chermak–Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups of G. The least element of the Chermak–Delgado lattice of G is known as the… Expand
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In this note we describe the structure of finite groups G whose Chermak-Delgado lattice is the interval [G/Z(G)] = {H \in L(G) \mid Z(G)\leq H\leq G}.
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If G is a finite group with subgroup H, then the Chermak–Delgado measure of H (in G) is defined as |H||C G (H)|. The Chermak–Delgado lattice of G, denoted ğ’žğ’Ÿ(G), is the set of all subgroups with… Expand
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Abstract Let G be a finite group and let H≤G. We refer to |H||CG(H)| as the Chermak–Delgado measure ofH with respect to G. Originally described by Chermak and Delgado, the collection of all subgroups… Expand
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