Character theory of finite groups has an important place in understanding the
structure of groups. In the literature, there are many publications from past
to present on the relationships between the structure of finite groups and their
complex irreducible characters. For example, Isaacs proved that a finite group
G is solvable whenever |cd(G)| ≤ 3, where cd(G) is the set of all irreducible
character degrees of G. Gagola and Lewis proved in [1] that a group G is nilpo-
tent if and only if χ(1)2 divides |G : kerχ| for every irreducible character χ of G.
Sometimes it is not necessary to study with the set of all irreducible characters
of a finite group to determine certain relations. Let G be a finite group and let χ
be an irreducible complex character of G. If G/kerχ has only one minimal nor-
mal subgroup then the irreducible character χ is called a monolithic character
of G. The literature contains some results using only monolithic characters. For
instance, Lu, Qin and Liu generalized in [2] that Gagola and Lewis’ theorem for
monolithic characters. In this talk, we generalize this theorem considering only
strongly monolithic characters. We also give some criteria for solvability and
nilpotency of finite groups by their strongly monolithic characters, which are a
subset of the set of monolithic characters of finite groups. The main theorems
has appeared in the article [3]. This is a joint work with Temha Erkoç. The
work in talk was supported by (TÜBİTAK). The project number is 119F295.
[1] Gagola S.M., Lewis M.L.: A character theoretic condition characterizing
nilpotent groups. Communications in Algebra. 27, 1053-1056 (1999).
[2] Lu J., Qin X., Liu X.: Generalizing a theorem of Gagola and Lewis char-
acterizing nilpotent groups. Arch. Math. 108, 337339 (2017).
[3] Bozkurt Güngor S., Erkoç T.: Some criteria for solvability and nilpotency
of finite groups by strongly monolithic characters. Bulletin of the Australian
Mathematical Society. 108, 120-124 (2023).