R ENDICONTI
del
S EMINARIO M ATEMATICO
della
U NIVERSITÀ DI P ADOVA
D ANU ¸ ˘ T M ARCU
A note on the projective representations of finite groups
Rendiconti del Seminario Matematico della Università di Padova, tome 102 (1999), p. 23-27
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of Finite Groups.
D0103NU0163 MARCU (*)
ABSTRACT - In this paper, we describe
exactly
the greatest common divisor of thedegrees
ofprojective representations
of finite groups.1. Introduction.
All
representations
andcharacters,
studied in this paper, are takenover the
complex numbers,
and all considered groups are finite. For ba- sic definitionsconcerning projective representations,
see[1].
If G is agroup and a is a
cocycle
ofG,
we denoteby Proj ( G , a ) = f r 1,
r 2,...,~}
the set of irreducible
projective
characters of G withcocycle
a, where(of course) t
is the number ofa-regular conjugacy
classes ofG, being
called the
degree
of Also asnormal, M( G )
will denote the Schur multi-plier
ofG, [a]
thecohomology
class of a, and[ 1, G ]
thecohomology
class of the trivial
cocycle
of G.The main result of this paper
exactly
describes thegreatest
commondivisor of the
degrees
ofProj ( G , a).
2. Main result.
Our
main
result is thefollowing
THEOREM. Let Pl, P2, 9 ... , Pn be the
prime
divisorsof
withPic, P2 ,
... ,Pn corresponding Sylow pi-subgroups of G.
Let be a sub-(*) Indirizzo dell’A.: Str. Pasului 3, Sect. 2, 70241-Bucharest, Romania.
24
group
of Pi of
minimal index such that[aMi ]
=[ 1, G]. Then,
thegreat-
est common divisor
of
thedegrees of Proj ( G , a)
isequal
toWe start
by defining
and
It is obvious that if
[a]
=[1, G],
thenc( G , a)
=s( G , a)
= 1.Thus,
weare
only really
interested in non-trivialcocycles
of G.Now,
wequote
thefollowing
well-knownresult, which,
to myknown, belongs
to thefolklore
of thistopic (e.g.,
for thepart (a),
see[1, 6.2.6]).
LEMMA 1. Let a be a
cocycle of
G witho([a])
= e inM(G).
Then,
(a) e I c(G, a);
(b) if p is a prime
number suchthat p I c(G, a ),
0We note here that it is not
true,
ingeneral,
thatc( G , a)
= e , or in- deedthat,
if someinteger m
dividesc(G, a),
then 7~e. For, according
to[2],
there exists acocycle
a of G =24
witho([a])
=2,
butc( G , a)
= 4 . In otherwords,
thecorresponding
central extension of theelementary
abelian group
G,
anextraspecial
group of order32,
has anunique
ordi-nary irreducible nonlinear
character,
which is of adegree
4.Now,
we show that toanalyse c( G , a)
we should consider theprime
divisors of
o([a])
ands(P, a p)
for thecorresponding Sylow subgroups
Pof G.
PROPOSITION 1. Let c =
c(G, a). Then,
thep-th part of
c, cp, isequal
to
s(P, a p) for
P aSylow p-subgroup of
G.PROOF. Let P E
Sylp(G)
andProj Now,
let r e
Proj ( G , a)
such thatThen,
where the6/s
arenon-negative integers
so thatHence, s(P , a p ) ~ cp .
On the other
hand,
let y EProj (P, a P) be
such thaty( 1 )
=s(P, a p).
Then,
for somenon-negative integers
ai , and so,comparing
the
p-th parts
of thedegrees,
we obtainHence,
cp ~a P ).
Thus,
we are left with the task ofdescribing s(P, a p)
=c(P, a P )
forP E Sylp(G). However,
we shallactually
consider a moregeneral
situ-ation than this. Recall that re
Proj ( G , a)
is calledmonomiaL,
if it is in-duced from a
projective
character ofdegree
1 of asubgroup,
and G issaid to be a
PM-group
if all its irreducibleprojective
characters aremonomial.
PROPOSITION 2. Let M be a
subgroup of
Gof
minimal index such that[aM]
=[1, G]. Then,
we have:(c) s(G, a) = [G: M] if and onLy if
there exists a monomial char- acterTE Proj (G, a )
with= s( G , a).
PROOF. Let
z’ E Proj ( G , a )
such thatt’(1) = s(G , a),
and A EE
Proj (M, a M )
with = 1.Then,
for somenon-negative
integers
a2 , and so26
proving ( a).
Sincec( G , a) a),
we have that(b)
is immediate from(a).
Now,
suppose thatequality
holds in(1). Then,
we must have thatÀ Gis
irreducible.Conversely,
ifre Proj(G, a)
is monomial andr(I) =
= s(G, a), then, by definition,
there exists asubgroup
L of Gand 03BC
ee
Proj (L , a L )
with = 1 such = r.Obviously,
then[ aL ]
==
[ 1, G ],
from Lemma1 (a). Also, [ G : L]
=s( G , a) ~ [ G : M], by (a), and, hence, by hypothesis, [G : L]
=[G : M].
Of course,
equality
inProposition 2(c)
does occur when G is a PM-group
and,
inparticular,
when G issupersolvable (see [1, 6.5.11]).
However,
if G= A4, o([a])
=2,
thens(G, a)
=c(G, a)
=2,
butA4
hasno
subgroup
of index2,
so thatequality
does notalways
hold.The
proof
of the main theorem is nowyielded by
the above remarks inconjunction
withPropositions
1 and 2.We mention three
applications
of the above results.COROLLARY 1. Let L be a
cyclic subgroup of
G.Then, s(G, a) ~
~
[ G : L]
andc( G , a) [ G : L ] for
allcocycles
aof
G.PROOF. Since L is
cyclic, M(L)
istrivial, and, hence, [ a L ]
=[1, G]
for all
cocycles
a of G.Thus,
the result isimmediate,
fromProposition 2(a).
Now,
we show that aslightly
weaker version ofProposition 2(a) gives
an alternative
proof
for the final assertion of[1, 4.1.9].
COROLLARY 2. Let e denote the
exponent of M(G),
a be acocycle of
G with
o([a])
= e , and L be asubgroup of
G such that[aL]
=[ 1, G].
Then, e|[G: L].
Inparticular,
e divides the indexof
eachcyclic
sub-group
of
G.PROOF.
By
Lemmal(a)
andProposition 2(a),
we havee|c(G, a)|[G: L]. D
Finally,
thefollowing type
of result is useful inconstructing
the pro-jective representations
of agiven
group withspecified Sylow
struc-ture.
COROLLARY 3. Let a be a
cocycle of
G with2 o( [ a ] ),
and supposethat G has a dihedral
Sylow 2-subgroup. The?4 (c(G, a ) )2
= 2 .PROOF. Let P E
Syl2 ( G ).
The restrictionmapping
fromSyl (M(G) )
into
M(P)
is amonomorphism. Hence,
since 1:J has acyclic subgroup
ofindex
2,
wehave, by Proposition
1 andCorollary 1,
that(c( G , a ) )2 =
= s(P, a p) = 2..
Acknowledgement.
I wish to thank the referee for his kindliness and interestconcerning
this paper.REFERENCES
[1] G. KARPILOVSKY,
Projective Representations of
FiniteGroups, Monographs
and Textbooks in Pure and
Applied
Mathematics, 94, Marcel Dekker (1985).[2] A. O. MORRIS,
Projective representations of
abelian groups, J. London Math.Soc., 7 (1973), pp. 235-238.
Manoscritto pervenuto in redazione il 4