Graph Invariants of Finite Groups via a Theorem of Lagarias

2006/12/20 by Fusun Akman, Akman, Fusun
Mathematics · #11R32 #20B05 #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT) #math.GR #math.NT #msc:11R32 #msc:20B05

paper · pdf · doi:10.48550/arxiv.math/0612618

LaTeX, 24 pages

arxiv created 2006/12/20 · arxiv updated 2009/12/01

Abstract

We introduce a new graph invariant of finite groups that provides a complete characterization of the splitting types of unramified prime ideals in normal number field extensions entirely in terms of the Galois group. In particular, each connected component corresponds to a division (Abteilung) of the group. We compute the divisions of the alternating group, and compile a list of characteristics of groups that the invariant reveals. We conjecture that the invariant distinguishes finite groups. Our exposition borrows elements from graph theory, group theory, and algebraic number theory.

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