R ENDICONTI
del
S EMINARIO M ATEMATICO
della
U NIVERSITÀ DI P ADOVA
M. J. I RANZO
M. T ORRES
The p
∗p-injectors of a finite group
Rendiconti del Seminario Matematico della Università di Padova, tome 82 (1989), p. 233-237
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The p*p-Injectors Group.
M. J. IRANZO - M. TORRES
(*)
1. Introduction and notation.
All groups considered in this note are finite. In
([2],
5.4Satz)
Blessenohl and Laue gave the
description
of the%*-injectors
of agrou
G,
where ~* is the class of thegeneralized nilpotent
groups;these
injectors
are thesubgroups VF*(G),
whereF*(G)
is the gen- eralizedFitting subgroup
of G and V describes theconjugacy
classof the
nilpotent injectors
of thesubgroup
=Oo(E(G))
where
E(G)
is thesemisimple
radical of G. In([4], Proposition 8)
it is shown that if
a
is aFitting
class between %(i.e.
the class ofnilpotent groups)
and%*,
then in a group G such thatF*(G) e a
there is a
unique conjugacy
class ofa-injectors
that areprecisely
the
R*-injectors
ofG,
andconversely.
The aim of this note is to
give
adescription
of thep*-by-p injec-
tors and the
p*p-injectors
of a group,by following
theanalogies
be-tween the class of
semisimple
groups and its radicalE(G)
and theclass of
p*-groups
and its radical0,.(G) (i.e.
thegeneralized p’-core
of
G)
and those between the class ofgeneralized nilpotent
groups(*)
Indirizzodegli
AA.: M. J. IRANZO:Departamento
deAlgebra,
Facul-tad de
Matemáticas,
Universidad de Valencia, 46100Burjasot (Valencia), Spain;
M. TORRES :Departamento
de Matemáticas, Universidad deZaragoza,
50009
Zaragoza, Spain.
Work
supported by
the DGICYT of theSpanish Ministry
of Education and Science,project
PS 87-0055-C02.234
and its radical
F*(G)
and the class of thep~‘p-groups
and its corres-ponding
radical as wasdeveloped by
Bender in[1].
Given a fixed
prime
p we shall denoteby e1)
the class of allp-groups,
(f1)’
the class of allp’-groups,
that of allp*-groups
andthat of the
p*p-groups ;
thecorresponding
radicals in a group Gare
denoted,
asusual, by 0~(G), 0~*(G)
andOp(G)
isthe
p-residual
of G.Finally
if H is asubgroup
ofG, C~(..g)
is thegeneralized
centralizer of g in G.For all definitions we refer to Bender
[1].
2. The
injectors.
LEMMA 1. The class
6p*@p - (G: OP(G)
E(t1).)
is aFitting
classcontaining
theFitting
class(t1).p.
Thecorresponding
radical isPROOF. The fact that is closed for normal
subgroups
isprecisely ([3],
Ch.X, 14.11).
If such that andNi
E i =1, 2,
we have and P nNi
isa
Sylow p-subgroup
ofNi , i
=1, 2,
if P is such asubgroup
ofG,
then
Ni = i =1, 2,
andfinally
we haveFor the
remaining
statements one can see([1], 4.13)
and([1],
6.1(iii), (iv)).
REMARKS.
1)
The inclusion in the lemma is strict for everyprime p,
take for instance E asimple
group withselfcentralizing Sylow p-subgroup
P andC,
thecyclic
group of order p, then theregular
wreathproduct
G = E wrC,
does notbelong
to(tp.p
because(here
Ep is the base group of G and Pp is aSylow p-sub-
group of
2)
The class of all groups that are a centralproduct
of ap*-
group and a p-group is not a
Fitting
class.LEMMA 2.
If a
is anyFitting
class such that theni)
Thea-maximal subgroups
of Gcontaining Gfj
are the sub-groups
(0,.(G)P)g
where P describes theSylow p-subgroups
of G.ii)
If then there is aSylow p-subgroup Po
of g such that
PROOF,
i)
Consider then0~*~(G) c S
allows us to use([1], 4.22)
and deduce thatOD*(~’)
=0~*(G) ;
butif, moreover, S
be-longs
toa,
we have S =0~* (G) Q
whereQ
is aSylow p-subgroup
of S.
By taking
aSylow p-subgroup
P of G such that we0~(~)~~~0~(~)jPy
so if S isif-maximal in G, clearly
S =
(0~*(G) P) ~ .
ii)
As before =O1’.(G),
so if T is aSylow p-subgroup
of and
Po
is one of .Hcontaining
T we can writetherefore
both
F-subgroups
ofG,
soby (i),
theequality
is valid.The first statement of this lemma means that the
Fitting
classesbetween and are dominant in
the
class of all finite groups,so it describes the
injectors
of every group(see [2],
5.1Satz).
Onthe other
hand, by
the secondpart
of the lemma everyinjector
ofa
group G
is aninjector
of eachsubgroup containing
it.REMARK. A group G is
p-constrained
if andonly
ifO9*(G) _
== OJ)I(G) (see [1],
6.12 or[3],
Ch.X,
14.12 Theoremb)~ ;
so if werestrict ourselves to the
Fitting
class of allp-constrained
groups wecan deduce from Lemma 2 that the class of
p-nilpotent
groups is dominant in the class ofp-constrained
groupsby obtaining
adescription
forthe p-nilpotent injectors
of ap-constrained
group Gthey
are thesubgroups
of the form where P describes the set ofSylow p-subgroups
of G and such asubgroup
is ap-nilpotent injector
of everysubgroup
of Gcontaining
it.But
conversely,
let us assume that G is a grouphaving
aunique conjugacy
class ofp-nilpotent injectors,
then the same is true forG
= thereforeby [5] E(G)
isp-nilpotent
and due to thefact that is
trivial, E(G)
must be trivial too i.e.G
is p-con-strained,
so G itself isp-constrained.
LEMMA 3. Let G be an and
Q
aSylow p-subgroup
of
0,.(G), then
=Op*(G)T
where T is aSylow p-subgroup
ofC*G(Q).
236
PROOF. Let us write X= Y = a
Sylow p-snb-
group of
0~*~(Ca(Q)),
and T aSylow p-subgroup
of0,,*,(Q)
such thatZ c T.
Put S = YT and observe that Y andS/ Y
arep*p-groups ;
on the other hand
S ~ YC*(Q) implies
so ~S = and
by ([1], 4.10)
we have that S is ap*p-group.
But on the other hand
by using ( [1 ], 4.18)
and([1 ], 6.10)
or([3],
Ch.
X, 14.13)
we can writeBut G is an so S = X.
From the last two lemmas we can assure
THEOREM. The
Ep*p-injectors
of agroup G
are thesubgroups
ofthe
conjugacy
classOJ).(G)T
where T describes the set of allSylow p-sybgroups
of where P describes the set of suchsubgroups
of G
and Q
those of0~*(G).
If one follows the notation and the unified
approach given by
Bender
([1], § 4), namely taking
for 7l either the set of allprimes
or a set
consisting
of asingle
one we cangive
a unified formula for the%*-injectors
and theEp*p-injectors
ofG;
in anabridged
formwhere if X is any group
X~
stands for aSylow n-subgroup
of .X inthe above
explained
sense.The
p*p-counterpart
of([4], Proposition 8)
is thefollowing
PROPOSITION.
Let F
be aFitting
class suchthat F
s andaS,
=a,
then if G is a group suchthat belongs
toa,
thenG has a
unique conjugacy
class ofa-injectors, namely
theinjectors. Conversely,
if G is a group whoseEp*Gp-injectors belong
to
~
then Ea.
PROOF. It is easy to prove that each
Ep*G*-injector
of Gbelongs to ~ ;
onthe
other handgiven
anya-injector H
ofG,
then H con-tains Gif
= so there is an V of Gcontaining H, therefore
.H =Y ; finally
use the dominance to deduce theexistence of such
injectors.
The converse is trivial.REFERENCES
[1] H. BENDER, On the normal p-structure
of
afinite
group and relatedtopics
I, Hokkaido Mathematical Journal, 7 (1978), pp. 271-288.[2] D. BLESSENOHL - H. LAUE,
Fittingklassen
endlicherGruppen,
in denengewisse Hauptfaktoren einfach
sind, J.Algebra,
56 (1979), pp. 516-532.[3] B. HUPPERT - N. BLACKBURN, Finite
Groups
III,Springer-Verlag (1982).
[4] M. J. IRANZO - F. PÉREZ MONASOR, &-constraint with
respect
to aFitting
class, Arch. Math., 46(1986),
pp. 205-210.[5] M. J. IRANZO - F. PÉREZ MONASOR, On the existence
of
certaininjectors
in
finite
groups, Comm. inAlgebra,
15 (1987), pp. 2193-2197.Manoscritto