Degenerations and order of graphs realized by finite abelian groups

TL;DR

This paper explores the concept of degeneration in graphs and groups, particularly finite abelian groups, and introduces a combinatorial framework to represent and analyze these degenerations using Young diagrams and partial orders.

Contribution

It establishes that degeneration in groups is a special case of graph degeneration, and provides a pictorial, combinatorial approach to classify finite abelian groups via saturated chains of Young diagrams.

Findings

Degeneration in groups is a specific case of graph degeneration.

Finite abelian p-groups can be represented by saturated chains of Young diagrams.

A combinatorial formula links the degree of projective representations to the structure of these chains.

Abstract

Let G_1 and G_2 be two groups. If a group homomorphism \varphi : G_1 \longrightarrow G_2 maps a \in G_1 into b \in G_2 such that \varphi(a) = b, then we say a degenerates to b and if every element of G_1 degenerates to elements in G_2, then we say G_1 degenerates to G_2. In this paper, we study degeneration in graphs and show that degeneration in groups is a particular case of degeneration in graphs. We exhibit some interesting properties of degeneration in graphs. We use this concept to present a pictorial representation of graphs realized by finite abelian groups. We discus some partial orders on the set T_p_1 \dots T_p_n of all graphs realized by finite abelian p_r-groups, where each p_r, 1 \leq r \leq n, is a prime number. We show that each finite abelian p_r-group of rank n can be identified with saturated chains of Young diagrams in the poset T_p_1 \dots T_p_n. We present a combinatorial formula which represents the degree of a projective representation of a symmetric group. This formula determines the number of different saturated chains in T_p_1 \cdots T_p_n and the number of finite abelian groups of different orders.