Contemporary Mathematics
Volume 402, 2006
The History of the Classification of Finite Groups with a
CC-Subgroup
Z. Arad and W. Herfort
Dedicated to Professor Marcel Herzog on the occasion of his retirement.
ABSTRACT. A proper subgroup M of a group G is called a CC-subgroup of G
if the centralizer Ca ( m) of every m E M# = M \ { 1} is contained in M. In
this paper, we classify all finite groups containing a CC-subgroup, extending
work of many authors.
1.
Definitions, Notation and Preliminaries
G
is a finite group.
1r(G) is the set of all primes dividing the order IGI.
For 1r
~
1r( G), let
7r1
denote 1r( G) \ 1r.
E1r: Following P. Hall, 1956
[26],
G satisfies E1f if G has a Hall 1r-subgroup.
C1f:
G
satisfies E1f and any two Hall 1r-subgroups are conjugate.
D1r:
G
satisfies C1f and every 1r-subgroup of
G
is contained in some Hall1r-subgroup.
E!;: G satisfies D1f and there exists a nilpotent Hall 1r-subgroup of G.
G is 1r-homogeneous: For every 1r-subgroup H:::; G, Nc(H)/Cc(H) is a 1r-subgroup.
Normal 1r-complement: A normal subgroup N
l
G is called a normal 1r-complement
if 1r(N)
~
1r' and 1r(GjN)
=
1r.
It is well-known that for 1r
=
{p},
p
a prime,
G
is p-homogeneous iff
G
has
normal p-complement
[23].
PROPOSITION.
(Arad and Chillag, 1984
[6])
Let 1r be a set of odd primes. Then
G
is 1r-homogeneous iff
G
has a normal 1r-complement.
It is well-known that if
M
is a CC-subgroup of
G,
then
M
is a Hall1r-subgroup
of
G
for 1r
=
1r(M).
Prime graph ofG (Gruenberg-Kegel, 1957
[24]):
Vertices are 1r(G). p,q
E
1r(G) is
joined by an edge iff G contains
a E
G of order pq.
2000
Mathematics Subject Classification.
Primary 20D25; Secondary 20D40.
Key words and phrases.
CC-subgroups, prime graph.
The second author would like to thank Netanya Academic College and Bar-Ilan University
for their generous hospitality in March 2004.
@2006 Z. Arad, W. Herfort
http://dx.doi.org/10.1090/conm/402/07569
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