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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 Chermak-Delgado lattice is a subgroup lattice of an elementary abelian $p$-group. We…

We investigate the question of how many subgroups of a finite group are not in its Chermak-Delgado lattice. The Chermak-Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasol\u{a} and…

Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by ${\cal CD}(G)$ the Chermak-Delgado lattice of $G$. In this note, we determine the finite groups $G$ such that $|{\cal CD}(G)|=|L(G)|-k$, $k=1,2$.

For a finite group G, we denote by v(G) the number of conjugacy classes of subgroups of G not in CD(G). In this paper, we determine the finite groups G such that v(G)=1,2,3.

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 Chermak-Delgado measure form a sublattice,…

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 Chermak-Delgado subgroup of $G$. This paper…

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}.

A group $G$ is said to have dense ${\cal CD}$-subgroups if each non-empty open interval of the subgroup lattice $L(G)$ contains a subgroup in the Chermak--Delgado lattice ${\cal CD}(G)$. In this note, we study finite groups satisfying this…

In a finite group G with subgroup H, the Chermak-Delgado measure of H (in G) is defined as the product of the order of H with the order of the centralizer of H. The Chermak-Delgado lattice of G, denoted CD(G), is the set of all subgroups…

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…

By imposing conditions upon the index of a self-centralizing subgroup of a group, and upon the index of the center of the group, we are able to classify the Chermak-Delgado lattice of the group. This is our main result. We use this result…

Let G be a finite group and let H be a subgroup of G. The Chermak-Delgado measure of H with respect to G is the product of the order of H with the order of the centralizer of H. Originally described by A. Chermak and A. Delgado, the…

In this note, we study the finite groups whose Chermak-Delgado measure has exactly two values. They determine an interesting class of $p$-groups containing cyclic groups of prime order and extraspecial $p$-groups.

Let $G$ be a finite group and $H\leq G$. The Chermak-Delgado measure of $H$ is defined as the number $|H|\cdot|C_{G}(H)|$. In this paper, we identify finite groups that exhibit the maximum number of Chermak-Delgado measures under some…

It is an open question in the study of Chermak-Delgado lattices precisely which finite groups $G$ have the property that $CD(G)$ is a chain of length $0$. In this note, we determine two classes of groups with this property. We prove that if…

The Chermak-Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups. We prove that the Chermak-Delgado lattice of a central product contains the product of the Chermak-Delgado lattices of the…

In this short note, we describe finite groups all of whose non-trivial cyclic subgroups have the same Chermak-Delgado measure.

In this paper, we completely classify the finite $p$-groups $G$ such that $\Phi(G')G_3\le C_p^2$, $\Phi(G')G_3\le Z(G)$ and $G/\Phi(G')G_3$ is minimal non-abelian. This paper is a part of the classification of finite $p$-groups with a…

Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by ${\rm Isolated}(G)$ the set of isolated subgroups of $G$. In this note, we describe finite groups $G$ such that $|{\rm Isolated}(G)|=|L(G)|-k$, where…

A group $G$ is said to have dense solitary subgroups if each non-empty open interval in its subgroup lattice $L(G)$ contains a solitary subgroup. In this short note, we find all finite groups satisfying this property.