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Enumerating typical abelian coverings of Cayley graphs

Haimiao Chen

TL;DR

The work of enumerating typical abelian coverings of Cayley graphs is complete by reducing the problem to enumerating certain subgroups of finite abelIAN groups.

Abstract

In this article we complete the work of enumerating typical abelian coverings of Cayley graphs, by reducing the problem to enumerating certain subgroups of finite abelian groups.

Enumerating typical abelian coverings of Cayley graphs

TL;DR

The work of enumerating typical abelian coverings of Cayley graphs is complete by reducing the problem to enumerating certain subgroups of finite abelIAN groups.

Abstract

In this article we complete the work of enumerating typical abelian coverings of Cayley graphs, by reducing the problem to enumerating certain subgroups of finite abelian groups.

Paper Structure

This paper contains 7 sections, 10 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Classifying typical abelian coverings of Cayley graphs
  3. Enumerating subgroups of finite abelian groups
  4. Overview
  5. On the structure of subgroups of abelian $p$-groups
  6. Subgroups with prescribed quotients
  7. Formulas for enumerating coverings

Key Result

Lemma 2.1

Two connected typical coverings $f_{i}:\emph{Cay}(A_{i},X_{i})\rightarrow\emph{Cay}(B,Y),\\i=1,2$, are isomorphic if and only if there exists a group isomorphism $\phi:A_{1}\rightarrow A_{2}$ such that $f_{2}\circ\phi=f_{1}$ and $\phi(X_{1})=X_{2}$.

Theorems & Definitions (23)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Remark 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • ...and 13 more