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On Posets of Classes of Subgroups with Same Set of Orders of Elements

Sachin Ballal, Tushar Halder

Abstract

In this paper, we study the posets of classes of subgroups of finite group having same set of orders of elements. We show that this poset is a chain only in the case of p-groups and moreover, we characterize all finite groups for which this poset is C2, the chain with two elements. We also show that this poset forms a lattice in the case of finite cyclic and dihedral groups and give a characterization when this lattice is distributive and modular.

On Posets of Classes of Subgroups with Same Set of Orders of Elements

Abstract

In this paper, we study the posets of classes of subgroups of finite group having same set of orders of elements. We show that this poset is a chain only in the case of p-groups and moreover, we characterize all finite groups for which this poset is C2, the chain with two elements. We also show that this poset forms a lattice in the case of finite cyclic and dihedral groups and give a characterization when this lattice is distributive and modular.
Paper Structure (2 sections, 12 theorems, 23 equations, 3 figures)

This paper contains 2 sections, 12 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Poset $\mathbf{\widetilde{\Pi}(\emph{G})}$

Key Result

Theorem 1.1

conrad Every subgroup of $D_n$ is cyclic or dihedral. A complete listing of the subgroups is as follows: Every subgroup of $D_n$ occurs exactly once in this listing.

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1
  • Theorem 2.1
  • proof
  • Corollary 2.1.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • ...and 13 more