Affiliation:
1. Dipartimento di Matematica , Universita degli Studi di Milano , Via Saldini 50, 20133 Milano , Italy
2. Centre for the Mathematics of Symmetry and Computation , University of Western Australia , 35 Stirling Highway , Perth 6009 , Australia
Abstract
Abstract
Let
cs
(
G
)
{\mathrm{cs}(G)}
denote the set of conjugacy class sizes of a group G, and let
cs
*
(
G
)
=
cs
(
G
)
∖
{
1
}
\mathrm{cs}^{*}(G)=\mathrm{cs}(G)\setminus\{1\}
be the sizes of non-central classes.
We prove three results.
We classify all finite groups for which (1)
cs
(
G
)
=
{
a
,
a
+
d
,
…
,
a
+
r
d
}
{\mathrm{cs}(G)=\{a,a+d,\dots,a+rd\}}
is an arithmetic progression with
r
⩾
2
{r\geqslant 2}
; (2)
cs
*
(
G
)
=
{
2
,
4
,
6
}
{\mathrm{cs}^{*}(G)=\{2,4,6\}}
is the smallest case where
cs
*
(
G
)
{\mathrm{cs}^{*}(G)}
is an arithmetic progression of length more than 2 (our most substantial result); (3) the largest two members of
cs
*
(
G
)
{\mathrm{cs}^{*}(G)}
are coprime.
For (3), it is not obvious, but it is true that
cs
*
(
G
)
{\mathrm{cs}^{*}(G)}
has two elements, and so is an arithmetic progression.
Funder
Australian Research Council
Engineering and Physical Sciences Research Council
Subject
Algebra and Number Theory
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Cited by
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