Graph Invariants of Finite Groups via a Theorem of Lagarias

Details Graph Invariants of Finite Groups via a Theorem of Lagarias Graph Invariants of Finite Groups via a Theorem of Lagarias

2006-12-20

arxiv.org/abs/math/0612618v1

Mathematics

Number Theory

LaTeX, 24 pages

Scientific paper

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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