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