DOI: 10.1515/jgth-2023-0263 ISSN: 1433-5883
Conjugacy class numbers and nilpotent subgroups of finite groups
Hongfei Pan, Shuqin Dong Abstract
Let 𝐺 be a finite group,
k
(
G
)
k(G)
the number of conjugacy classes of 𝐺, and 𝐵 a nilpotent subgroup of 𝐺.
In this paper, we prove that
|
B
O
π
(
G
)
/
O
π
(
G
)
|
≤
|
G
|
/
k
(
G
)
\lvert BO_{\pi}(G)/O_{\pi}(G)\rvert\leq\lvert G\rvert/k(G)
if 𝐺 is solvable and that
15
7
|
B
O
π
(
G
)
/
O
π
(
G
)
|
≤
|
G
|
/
k
(
G
)
\frac{15}{7}\lvert BO_{\pi}(G)/O_{\pi}(G)\rvert\leq\lvert G\rvert/k(G)
if 𝐺 is nonsolvable, where
π
=
π
(
B
)
\pi=\pi(B)
is the set of prime divisors of
|
B
|
\lvert B\rvert
.
Both bounds are best possible.