Degenerations and order of graphs realized by finite abelian groups

Abstract

Let G 1 and G 2 be two groups. If there exists a homomorphism ϕ : G 1 → G 2 such that ϕ(a) = b, where a ∈ G 1 and b ∈ G 2 , then a group G 1 degenerates to a group 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 •••p n of all graphs realized by finite abelian p r -groups, where each p r , 1 ≤ r ≤ 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 •••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 •••p n and the number of finite abelian groups of different orders.