Degenerations and order of graphs realized by finite abelian groups

Rameez Raja

Published 2022-05-18Version 1

Let G_1 and G_2 be two groups. If there exists a homomorphism \phi from G_1 to G_2 such that \phi(a) = b, 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 \cdots 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 \cdots 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 p_n and the number of finite abelian groups of different orders.