The Chermak-Delgado Measure as a Map on Posets

Abstract

The Chermak-Delgado measure of a finite group is a function which assigns to each subgroup a positive integer. In this paper, we give necessary and sufficient conditions for when the Chermak-Delgado measure of a group is actually a map of posets, i.e., a monotone function from the subgroup lattice to the positive integers. We also investigate when the Chermak-Delgado measure, restricted to the centralizers, is increasing.

Figures (3)

• Figure 1: A diagram for $G=S_3$ showing the Chermak--Delgado measure $m_G:\mathcal{S}(G)\rightarrow \mathbb{Z}^+$. All of the subgroups isomorphic to $C_2$ have measure $4$.
• Figure 2: A diagram for $G=D_8$ showing the Chermak--Delgado measure $m_G:\mathcal{S}(G)\rightarrow \mathbb{Z}^+$. The two regions of the diagram are the fibers of $m_G$. The map $m_G$ is a monotone map of posets as well.
• Figure 3: A diagram for $G=A_4$ showing the Chermak--Delgado measure $m_G:\mathcal{S}(G)\rightarrow \mathbb{Z}^+$. We note that each copy of $C_3$ has measure $9$ and all of the copies of $C_2$ have measure 8. Since $1$ has measure 12, the map $m_G$ is not a monotone map of posets.

Theorems & Definitions (39)

• Theorem A
• Theorem B
• Theorem C
• Proposition 1
• Proposition 2: FGT
• Proposition 3
• proof
• Corollary 4
• Proposition 5: Wielandt FGT
• Proposition 6