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LİSANSÜSTÜ ARAŞTIRMALAR SEMPOZYUMU & TANITIM GÜNLERİ 2018

On Character Degrees of Finite Groups and Some Associated Graphs

Nour ALNAJJARİNE, Roghayeh HAFEZİEH
Matematik ABD

ÖZET
The character degree set of a finite group G, cd(G), is defined to be the set of all character degrees of G, that

is, cd(G)={χ(1) : χ ∈ Irr(G)}, where Irr(G) is the set of all complex irreducible character degrees of G. It is well-known
that there is a strong interplay between the arithmetical properties of cd(G) and the structure of the group G. For
instance, the celebrated Ito-Michler's Theorem proves that if a prime p divides no character degree of G, then G has
a normal abelian Sylow p-subgroup.

While studying the character degree set of a finite group G, it is useful to attach some graph structure on
cd(G), such as Δ(G), Γ(G) and B(G), which are strongly related undirected graphs [1]. It has been always an interesting
question to know which graphs can occur as Δ(G), Γ(G) or B(G), for some finite groups G. LiGuo and GuoHua, for
example, gave a classification of all four-vertex graphs that occur as Γ(G) for a nonsolvable group G [2]. On the other
hand, M. Lewis and D. White claimed in [3] that there is no nonsolvable group G such that Δ(G) is a P_3 or a C_4.
Another interesting result is due to Tong-Viet who showed that if G is a finite group such that Δ(G) has no triangles,
then Δ(G) has at most 5 vertices. Furthermore, he classified all five-vertex graphs that have no triangles and can
occur as Δ(G) for some finite group G [4].

The aim of this thesis is to investigate the relationship between the structure of finite group G and the graphs
associated with its character degree. In particular, we discuss properties of the bipartite divisor graph B(G) with the
focus on the cases where B(G) is a path, union of paths or a cycle.
Anahtar Kelimeler: Character Degree Set, Nonsolvable Group, Path.

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