Affiliation:
1. School of Mathematics and Statistics, Southwest University , Chongqing 400715 , P. R. China
2. School of Mathematics and Statistics, Chongqing Technology and Business University , Chongqing 400067 , P. R. China
Abstract
Abstract
The aim of this article is to study how the greatest common divisor of the degree and codegree of an irreducible character of a finite group influences its structure. We study a finite group
G
G
with
gcd
(
χ
(
1
)
,
χ
c
(
1
)
)
{\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1))
a prime for almost all irreducible characters
χ
\chi
of
G
G
, and obtain the following two conclusions:
(1)
There does not exist any finite group
G
G
such that
gcd
(
χ
(
1
)
,
χ
c
(
1
)
)
{\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1))
is a prime, for each
χ
∈
Irr
(
G
)
♯
\chi \in {\rm{Irr}}{\left(G)}^{\sharp }
, where
Irr
(
G
)
♯
{\rm{Irr}}{\left(G)}^{\sharp }
is the set of non-principal irreducible characters of
G
G
.
(2)
Let
G
G
be a finite group, if
gcd
(
χ
(
1
)
,
χ
c
(
1
)
)
{\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1))
is a prime, for each
χ
∈
Irr
(
G
)
\
Lin
(
G
)
\chi \left\in {\rm{Irr}}\left(G)\backslash {\rm{Lin}}\left(G)
, then
G
G
is solvable, where
Lin
(
G
)
{\rm{Lin}}\left(G)
is the set of all linear irreducible characters of
G
G
.