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Structure Of Finite, Minimal Nonabelian Groups And ...

STRUCTURE OF FINITE, MINIMAL NONABELIAN GROUPS
AND TRIANGULARIZATION
MITJA MASTNAK AND HEYDAR RADJAVI
To Tom La ey on his sixty- fth birthday
Abstract. Motivated by problems concerning simultaneous triangularization,
we study the structure of nite, minimal nonabelian groups. Using the structure
result of Miller and Moreno we explicitly describe all irreducible representations
of such groups. We illustrate the usefulness of results of this type on several
examples.
HF Introduction
gert—in nite su˜groups of m—tri™es h—ve — signi™—nt role in proving redu™i˜ility
—nd tri—ngul—riz—˜ility results for semigroups of oper—torsF fy reducibility of —
™olle™tion ƒ of ˜ounded line—r oper—tors on — f—n—™h sp—™e ˆ @of nite or innite
dimensionAD we me—n the existen™e of — nontrivi—l ™losed su˜sp—™e of ˆ D inv—ri—nt
under @every oper—tor inA ƒF fy @simult—neousA triangularizability of ƒ is me—nt the
existen™e of — m—xim—l ™h—in of su˜sp—™es of ˆ e—™h mem˜er of whi™h is inv—ri—nt
under ƒF yf p—rti™ul—r interest to us is the ™—se in whi™h ƒ is — semigroupD iFeFD
is ™losed under multipli™—tionF et le—st when ƒ ™ont—ins some nonzero ™omp—™t
oper—tor @or in p—rti™ul—r when ˆ is niteEdimension—lAD its redu™i˜ility is often
determined ˜y th—t of nite m—trix groups —sso™i—ted with itF „hese —re minim—l
non—˜eli—n groups of — spe™i—l typeD —nd it is useful to know —s mu™h —s possi˜le
—˜out their stru™ture —nd represent—tionsF
„hey —re —ll solv—˜le groupsD of ™ourseD —nd mu™h of the stru™ture results preE
sented ˜elowD espe™i—lly —˜out the nilpotent ™—seD ™—n ˜e o˜t—ined from —˜str—™t
group theory ‘R“F
„he relev—nt p—rts of our resultsD eFgFD „heorems PFPFTD PFQFPD —nd goroll—ry PFQFQD
@—lthough e—sily dedu™ed from ˜—si™ group represent—tion theoryA seem to ˜e newF
yne of our m—in ™ontri˜utions is unied not—tion whi™h m—kes the results more
—™™esi˜le to oper—tor theorists —nd e—sier to use in the ™ontext of simult—neous
tri—ngul—riz—˜ilityF
…sing the stru™ture results we will provide new —nswers to questions of the folE
lowing formF vet f ˜e — homogeneous polynomi—l in two non™ommuting v—ri—˜lesF
‡h—t ™onditions ™—n ˜e imposed on f so th—t wheneverD for —ll A; B in — semiE
groupD f@A; BA is ’sm—ll4 in some sense @eFgF is zeroD nilpotentD qu—sinilpotentD
1
2
MITJA MASTNAK AND HEYDAR RADJAVI
et™FAD then the semigroup is tri—ngul—riz—˜leD or —t le—st redu™i˜leF ‡e ˜elieve th—t
our results on the stru™ture of these m—trix groups will nd further —ppli™—tionsD
in™luding simpli™—tions of known tri—ngul—riz—˜ility theoremsF
IF Notation
sf q is — power of — primeD then Fq denotes the eld with q elementsF sf m is —
positive integerD then gm denotes the ™y™li™ group of order m —nd  m denotes the
m ¢ m –™y™le m—trix9D iFeFD
HH I
I
f
FF
g
 
f
F FFF
g
m a d
H Ie :
I
H
sf G is — groupD then we denote the su˜set of its nEth powers ˜y Gn a fgnj g P GgF
‡e denote group ™ommut—tors ˜y ‘ ;  “ —nd ring ™ommut—tors ˜y ‘ ;  “rD more
pre™iselyD if x; y P GD a; b P RD where G is — group —nd R is — ringD then ‘x; y“ a
xyx 1y 1 —nd ‘a; b“r a ab   baF ell represent—tions we ™onsider —re over ™omplex
num˜ersF
IFIF @p; qA-polynomials and @p; qA-matrices. sn our ™onsider—tions ™ert—in polyE
nomi—ls pl—y — very ™entr—l roleF
De nition 1.1.1. Let p; q be primes (not necessarily distinct). We say that a
monic polynomial f P Z‘x“ is a @p; qA-polynomial provided that:
@IA if p Ta q, then f modulo q is an irreducible factor of xp   I P Fq‘x“ distinct
from x   I,
@PA if p a q Ta P, then f@xA a @x   IA2,
@QA if p a q a P, then either f a H, or f@xA a x C I.
Remark 1.1.2. If p; q are distinct primes and f@xA a a0 C: : :Cam 1xm 1 Cxm Ta
x   I is an irreducible factor of xp   I P Fq‘x“, then the coecient a0 P Fq is
nonzero and for at least one choice of f , the coecient am 1 is invertible as well
€
(since I a
f am 1@f A).
‡e use @p; qAEpolynomi—ls to dene groups of di—gon—l m—tri™esF
De nition 1.1.3. If p; q are primes and f@xA a a0 C : : : am 1xm 1 C xm is a
@p; qA-polynomial, then we de ne a group of diagonal p ¢ p matrices h a h@p; qY fA
as follows:
@IA if @p; q; fA Ta @P; P; x C IA, then
¨
©
h a di—g @1; : : : ; pAj a0
j a1
j+1 ¡ : : : ¡ am 1
j+m 1j+m a I; qj a I ;
MINIMAL NONABELIAN GROUPS AND TRIANGULARIZATION
3
@PA if @p; q; fA a @P; P; x C IA, then
h a f¦I2; ¦ di—g @i;  iAg :
Remark 1.1.4. If pjq   I then f@xA a x    and
n


o
h@p; qY fA a di—g ; ; : : : ; p 1  q a I :
If the smallest integer m such that pjqm   I is p   I, then f@xA a I C x C : : : C xp 1
and h@p; qY fA consists of all order q diagonal matrices of determinant I.
If p Ta P, then
¨
 
¡
©
h@p; p; fA a jI; j di—g I; k; : : : ; (p 1)k  j; k a H; : : : ; p   I ;
where  is any primitive p-th root of I.
st will turn out th—t h@p; qY fA is inv—ri—nt under the ™onjug—tion ˜y  pF sn view
of this we deneX
De nition 1.1.5. If p; q are primes, H Ta P C, and f a @p; qA-polynomial, then
we de ne a group of p ¢ p matrices q a q@p; qY Y fA by
¨

©
q a D@   
pAk D P h@p; qY f A; I k pj :
y˜serve th—t if p a ID then q@p; qY Y fA 9 q@p; qY IY fAF
De nition 1.1.6. If p; q are primes and j a positive integer, then a group of
matrices q is said to be a @p; q; jAEm—trix group , if there exists a primitive pj-th
root of unity and a @p; qA-polynomial f (with f@xA a xCI, if @p; q; jA a @P; P; PA),
such that, up to simultaneous similarity, we have q a q@p; qY Y fA.
yne of the m—in results in this p—per is th—t every irredu™i˜leD niteD minim—l
non—˜eli—n m—trix group is — @p; q; jAEm—trix groupD —nd th—t for — xed triple
@p; q; jAD —ll @p; q; jAEm—trix groups —reD up to simil—rityD the s—meF
PF Structure Theory
piniteD minim—l non—˜eli—n groups were rst investig—ted ˜y willer —nd woreno
‘R“F „hey rst proved th—t they —re solv—˜le —nd then used solv—˜ility to o˜t—in
— ™omprehensive stru™ture result for su™h groups @see „heorems PFIFP —nd PFPFPAF
st should ˜e noted th—t yF tF ƒ™hmidt extended their solv—˜ility result to niteD
minim—l nonnilpotent groups ‘V“F
felow we dedu™e the stru™ture results of willer —nd worenoF por the s—ke of
™ompleteness —nd —lso to m—ke the results more —™™essi˜le to oper—tor theorists
we in™lude the proofsF yur —ppro—™h is slightly dierent from th—t of willer —nd
worenoD fo™using mostly on des™ri˜ing groups in terms of gener—tors —nd rel—tions
r—ther then exploring their —˜str—™t stru™ture @eFgF ™ounting the the num˜er of
ƒylow su˜groups of ™ert—in sizeAF „his en—˜les us to expli™itly des™ri˜e irredu™i˜le
represent—tions of these groupsF
4
MITJA MASTNAK AND HEYDAR RADJAVI
„hroughout this se™tionD G denotes — niteD minim—l non—˜eli—n groupF xote
th—t the ™ommut—tor su˜group ‘G; G“ of G is @due to solv—˜ilityA — proper su˜E
group of G —nd is thus —˜eli—nF elso o˜serve th—tD due to minim—lityD every p—ir
of non™ommuting elements gener—tes GF st is worth pointing out th—t — homomorE
phi™ im—ge of — minim—l non—˜eli—n group is either —˜eli—n or minim—l non—˜eli—nF
ren™e the r—nge of every nons™—l—r irredu™i˜le represent—tion of — minim—l nonE
—˜eli—n group is —lso — minim—l non—˜eli—n groupF
PFIF The nilpotent case. sf G is nilpotentD then note th—t G must ˜e — pEgroup
for some prime pF sndeedD one of ƒylow su˜groups of G must ˜e non—˜eli—n @sin™e
G is not —˜eli—n —nd is — dire™t produ™t of its ƒylow su˜groupsA —nd ˜y minim—lity
G is equ—l to th—t groupF
vet Gn ˜e the nEfold iter—ted ™ommut—tor of G @iFeFD G0 a G —nd Gn a ‘G; Gn 1“
for n ! IAF vet r ˜e minim—l su™h th—t Gr a IF xote th—t r ! P —s G is
not ™ommut—tiveF vet a P Gr 2 —nd b P G ˜e — p—ir of non™ommuting @hen™e
gener—tingA elements of GF xote th—t ‘a; b“ P Gr 1  Z@GA @—nd hen™e r a PAF
‡ithout loss we m—y —lso —ssume th—t ap; bp lie in Z@GA @™ontinue to repl—™e a ˜y
ap or b ˜y bp until this is soAF
‡e h—ve shown th—t G is gener—ted ˜y a of orderD s—y piD —nd b of order pj —nd
ap; bp; ‘a; b“ P Z@GAF ‡e —ssume th—t i C j is minim—l possi˜leF ƒin™e ‘a; b“ is —
™entr—l elementD every element of G ™—n ˜e written in the form ‘a; b“n1an2bn3F
sf ap is — pEth power of — ™entr—l elementD then ap a I @if ap a xpD where
x P Z@GAD then repl—™e a ˜y ax 1AF „he s—me holds for bpF xote —lso th—t
‘a; b“p a ‘ap; b“ a IF
„he stru™ture of G is thus determined up to the stru™ture of its ™entre Z@GA a
hap; bp; ‘a; b“i @if a ™ommutes with ‘a; b“n1an2bn3D then a must ™ommute with bn3 —nd
hen™e n3 must ˜e divisi˜le ˜y pAF
Lemma 2.1.1. If p Ta P, then hapi ’ hbpi a I. Furthermore, if p a P, then either
hapi ’ hbpi a I, or hapi a hbpi and i a j a P.
Proof. essume I Ta arpn a bspmD where r; s —re ™oprime to p —nd I n < i —nd
I m < jF ‡ith no loss of gener—lity we —ssume th—t r a s a I —nd th—t n mF
sf n < mD then we ™ould repl—™e a ˜y ab pm nD ™ontr—di™ting the minim—lity of iCjF
ren™e n a mF ƒin™e I a api a @apnApi n a @bpnApi n a bpiD we must h—ve i jF e
symmetri™ —rgument shows th—t j i —nd thus i a jF xote th—t ‘ab 1; b“ a ‘a; b“
  ¡
—nd @ab 1Apn a apnb pn‘a; b“@pnA
A
2
a ‘a; b“@pn2 F sf either n > ID or p Ta PD then pn2
is divisi˜le ˜y p —nd hen™e repl—™ing a ˜y ab 1 would ™ontr—di™t the minim—lity
of i C jF essume now th—t n a I —nd p a PF „hen ap a bpF elsoD if i > PD then
@ab 1Ap2 a I —nd repl—™ing a ˜y ab 1 would —g—in ™ontr—di™t the minim—lity of
i C jF
£
‡e m—y —ssume without loss of gener—lity th—t either hapi ’ hbpi a ID or ap a bp
with p a i a j a PF „he order of G is jZ@GAjp2F sf ap a bpD then the size of Z@GA
MINIMAL NONABELIAN GROUPS AND TRIANGULARIZATION
5
is either pi @if ‘a; b“ TP haiA or pi 1F sf hai ’ hbi a ID then the size of Z@GA is either
pi+j 1 @if ‘a; b“ TP ha; biAD or pi+j 2F
y˜serve th—t ‘G; G“ a h‘a; b“i is — ™y™li™ group of order p —nd th—t every element
of ‘G; G“ is — ™ommut—torF sndeedD ‘a; b“i a ‘ai; b“ a ‘a; bi“F
„he —˜ove dis™ussion is summ—rized in the proposition ˜elowF
Theorem 2.1.2 @™fF ‘R“A. If G is a nite, nilpotent, minimal nonabelian group,
then
@IA For some prime p, G is a p-group and is generated by a; b P G such that
ap; bp and ‘a; b“ are central elements.
@PA If p Ta P, then we can additionally assume that hapi ’ hbpi a I. If p a P,
then either ha2i ’ hb2i a I, or a2 a b2 and a4 a I a b4.
@QA ‘G; G“ a h‘a; b“i 9 Cp, and every element of ‘G; G“ is a commutator.
£
Corollary 2.1.3. If G is a nite, nilpotent, minimal nonabelian group with a cyclic
centre, then G is generated by a; b P G such that ap a I, and bp; ‘a; b“ P Z@GA.
Theorem 2.1.4. Any irreducible nonscalar representation of G has size p and is
given by a U3 A a A ;, b U3 B a B , where
 
¡
A a di—g I; ; : : : ; p 1 ; B a  p;
where  is a primitive p-th root of unity and pi a I a pj. If ap a bp, then we
have p a p. If ‘a; b“ P hap; bpi, say ‘a; b“ a ai0pbj0p, then we have  a i0p j0p.
Representations associated to @ ; ; A and @ H; H; HA are isomorphic if and only
if @ p; p; A a @ Hp; Hp; HA.
Proof. y˜serve th—t ‘A; B“ a  1ID Ap a pID Bp a pI —nd hen™e a U3 A —nd
b U3 B is indeed — represent—tion of GF st is ™le—rly irredu™i˜leF ƒuppose th—t
represent—tions —sso™i—ted to @ ; ; A —nd @ H; H; HA —re isomorphi™F „hen there
is —n inverti˜le m—trix C su™h th—t CA ;C 1 a A H;H —nd CB C 1 a B HF ren™e
H 1I a ‘A H;H; B H“ a C‘A ;; B “C 1 a C@ 1IAC 1 a  1I —nd therefore  a HF
elso HpI a Ap H a CAp C 1 a pI —nd HpI a Bp H a CBp C 1 a pIF ren™e
@ p; p; A a @ Hp; Hp; HAF
st is now su™ient to prove th—t represent—tions —sso™i—ted to @ ; ; A for p—irE
wise distin™t @ p; p; A together with the s™—l—r represent—tions @whi™h —re given
˜y ™h—r—™ters on G=‘G; G“ a G=h‘a; b“iA exh—ust —ll possi˜ilitiesF „his is done ˜y
using — ™ounting —rgumentF yne mustD —™™ording to the stru™ture of Z@GAD ™onsider
four ™—sesF rere is one of the ™—ses @the others work —lmost the s—me w—yAF esE
sume th—t ap Ta bp —nd th—t ‘a; b“ TP hap; bpiF sn this ™—se jGj a p2jZ@GAj a pi+j+1F
6
MITJA MASTNAK AND HEYDAR RADJAVI
ƒumm—tion of sizes of represent—tions des™ri˜ed —˜ove gives
ˆ
ˆ
ˆ
dim@A2 a
jf@ p; p; Agjp2 C
jG=‘G; G“jI2

a pi 1pj 1@p   IAp2 C pi+j+1 1 a pi+j+1 a jGj
£
PFPF The nonnilpotent case. essume th—t G is not nilpotentF vet a P ‘G; G“
—nd b P G gener—te GF xote th—t without —ny loss of gener—lity we ™—n —ssume
th—t aqi a I a bpjD where p —nd q —re primesD —nd i; j —re positive integersF ‡e
—ssume th—t j is sm—llest possi˜leD —nd hen™e bp ™ommutes with aF
‡e rst show th—t if p a qD then G is — qEgroup —nd therefore nilpotentF vet
H a hbrasb rj r; si ˜e the norm—l su˜group of G gener—ted ˜y aF „he group H
is —˜eli—n —s H ‘G; G“D —nd therefore it is — qEgroup sin™e it is gener—ted ˜y
elements of exponent — power of qF xote th—t every g P G ™—n ˜e written in the
form g a bnhD where h P HF y˜serve th—t @bnhAm a bnmhH for some hH P H —nd
hen™e Gpj a fgpjjg P Gg  HF „hus if p a qD then the exponent of G must divide
qi+jF
prom now on —ssume th—t p Ta q —nd o˜serve th—t hbi’H a IF ren™e G a Hohbi
@H o hbi is — non™ommut—tive su˜group of G —nd is therefore equ—l to GAF
‡e will show th—t i a ID iFeFD aq a IF hue to minim—lity of G we m—y —ssume
th—t H a ‘G; G“ a ‘H; G“ —nd th—t Hq a fxqj x P Hg is — ™entr—l su˜group of
GF xote th—t every nontrivi—l ™ommut—tor ‘x; y“ a xyx 1y 1D where x P HD h—s
exponent qF sndeedD sin™e x ™ommutes with ‘x; y“ we h—ve ‘x; y“q a ‘xq; y“ a IF
„hus a —nd hen™e —lso H h—ve exponent qF y˜serve —lso th—t bp P Z@GAF ren™e
G 9 @Frq; CAoCpjD where the —™tion of Cpj a hbi on Frq is given ˜y b@uA a BuD where
B P GLr@FqAD Bp a IF sf u P FrqD —nd bn P CpjD then we identify u a u o I P G
—nd bn a 0 o bn P GF
‡e next show th—t r a mD where m is the minim—l positive integer su™h th—t
qm   I is divisi˜le ˜y pF xote th—t the eld Fqm is the sm—llest eld extension of
Fq ™ont—ining — primitive pEth root of unityF ‚e™—ll th—t the q—lois group of this
extension is ™y™li™ —nd is gener—ted ˜y 'X Fqm 3 FqmD '@xA a xpF „he order of '
is m @—nd hen™e m divides p   IAF ‡e denote the indu™ed @entrywiseA —™tion on
Frq ˜y ' —s wellF y˜serve th—t the irredu™i˜le f—™tors of xp   I over Fq —re x   I
—nd f! a @x   !A@x   !qA : : : @x   !qm 1AF elso note th—t f! a f if —nd only if
1
!2
!1 —nd !2 —re in the s—me or˜it under the —™tion of h'i @iFeFD if !2 a !qi1AF
vet B X Frqm 3 Frqm ˜e the line—r tr—nsform—tion indu™ed ˜y B —nd let u P Frqm
˜e — nonzero eigenve™tor ™orresponding to —n eigenv—lue ! Ta I @su™h —n eigenv—lue
MINIMAL NONABELIAN GROUPS AND TRIANGULARIZATION
7
exists —s B Ta ID Bp a ID —nd charFqm a q does not divide pAF hene
H
I
I
I
: : :
I
f !
!q
: : :
!qm 1 g
 a f
f
g
d FF
F
F
g
F
F
F
F
F
e
!m 1 !(m 1)q : : : !(m 1)qm 1
—nd note th—t the m—trix
 
¡
X a u '@uA : : : 'm 1@uA 
h—s entries in Fq @they —re xed under 'A —nd th—t r—nk X a m @ is —n invertE
i˜le †—ndermonde m—trix —nd u; '@uA; : : : ; 'm 1@uA —re eigenve™tors ˜elonging to
distin™t eigenv—lues of BAF y˜serve th—t ColXD the ™olumn sp—™e of XD is —n inE
v—ri—nt su˜sp—™e for B —nd th—t @ColXAohbi is — non—˜eli—n su˜group of GF ren™e
ColX a Frq —nd r a mF y˜serve —lso th—t B is irredu™i˜leX if …  ColX a Frq is
—n inv—ri—nt su˜sp—™e for BD then B… Ta ID —s BColX h—s no nontrivi—l xed pointsD
—nd hen™e … o hBi is — non—˜eli—n su˜group of GF
vet f@xA a a0 C a1x C : : : C am 1xm 1 C xm ˜e the ™h—r—™teristi™ polynomi—l of
B @whi™h is —lso the minim—l polynomi—l —nd f a f! for some !AF xote th—t for
—ny nonzero ve™tor v0 P FmqD the ve™tors
v0; Bv0; : : : ; Bm 1v0
form — ˜—sis for Fmq —nd th—t
HH
 a I
0
fI H
 a g
B a B
f
1 g
f a d
FFF FFF
F
F
F e
I  am 1
with respe™t to this ˜—sisF
De nition 2.2.1. We write Gf a Gf;j a Fmq o hbjbpj a Ii, where the action of b
on Fmq is given by Bf.
y˜serve th—t if g@xA Ta x I is —nother irredu™i˜le divisor of xp ID then Gf 9 GgF
sf f —nd g —re —sso™i—ted to !1 —nd !2D where !2 a !i1D then the isomorphism is
indu™ed ˜y Bg U3 Big —nd u0 U3 v0F
essume now th—t G a GfF y˜serve th—t ‘G; G“ a FmqF ‡e will show th—t every
element of ‘G; G“ is — ™ommut—torF hene K a fbub 1u 1ju P Fmqg —nd note
th—t K is — norm—l su˜group of ‘G; G“F ƒin™e bub 1u 1 a @B   IAu —nd B   I is
inverti˜leD it follows th—t K a FmqF ‡e h—ve thus proven the following theoremF
Theorem 2.2.2 @™fF ‘R“A. If G is a nite, nonnilpotent, minimal nonabelian group,
then G 9 Gf for some irreducible divisor f of xp 1 P F
x 1
q‘x“. The size of G is qmpj.
8
MITJA MASTNAK AND HEYDAR RADJAVI
The commutator subgroup ‘G; G“ is isomorphic to Cm
q . Every element of ‘G; G“ is
a commutator.
£
PFPFQF Representations. rere we des™ri˜e the irredu™i˜le ™omplex represent—tions
of G a Gf;j a @Fmq; CA o CpjF ell su™h represent—tions —re o˜t—ined —s follows @™fF
‘SD €roposition PS“AF ghoose D — ™h—r—™ter on H a @Fmq; CA —nd D — ™h—r—™ter
on Cpj a hbpi 9 gppiF „he ™orresponding irredu™i˜le represent—tion of G is then
indu™ed ˜y   D — represent—tion of H o Cppi GF wore pre™iselyD if I Ta  P ˜
H
—nd P CD pi a ID then G —™ts on V a V @; AD — ve™tor sp—™e with ˜—sis
v0; : : : ; vp 1D ˜y hvi a @bihb 1Avi —nd bvi a vi+1F xote th—t represent—tions
V @; A —nd V @H; HA —re isomorphi™ if —nd only if H a bi@A for some integer
H i < p —nd p a HpF y˜serveD th—t if p a ID then without —ny loss of
gener—lity we m—y —ssume th—t a IF sf  a ID then we get the IEdimension—l
represent—tions given ˜y h U3 ID h P HD b U3 D pj a IF ‡e summ—rize the
dis™ussion —˜ove in the theorem ˜elowF
Theorem 2.2.4. Let G be a nonnilpotent, nite, minimal nonabelian group. Then
G a Gf;j a @Fmq; CAohbjbpj a Ii. If 0 Ta u P Fmq, then every nonscalar representa-
tion of G is determined by u U3 A and b U3 B, where A is an arbitrary nonidentity
element of h@p; qY fA and B a  p, with pj a I. Representations associated to
@A;  pA and @AH; H pA are isomorphic if and only if A a AH and p a Hp.
Proof. e˜˜revi—te ei a @H; : : : ; I; : : : ; HA P FmqF ‡ithout —ny loss of gener—lity
—ssume th—t u a e1F xow o˜serve th—t nontrivi—l ™h—r—™ters  dis™ussed —˜ove
—re in ˜ije™tive ™orresponden™e with h@p; qY fAF „his ™orresponden™e is given ˜y
A@ejA a jD where A a di—g @1; : : : ; pA P h@p; qY fAF
£
Corollary 2.2.5. If p Ta q, then the size of h a h@p; qY fA is qm. If @ 0; : : : ; m 1A
is an m-tuple of complex numbers of order q, then for every integer i, there exists a
unique element D a di—g @1; : : : ; pA of h such that i+k a k, for H k m   I.
£
„he following is result —˜out the frequen™y with whi™h entries in h@p; qY fA o™™ur
is of some interestF
Theorem 2.2.6. If f@xA Ta x I is an irreducible divisor of xp I P Fq and q a I
( can be I), then the expected number of 's in a random D P h a h@p; q; fA is
p . In particular, if q > p, then there are members of h containing no .
q
Proof. „he num˜er of —ll Hs in —ny xed di—gon—l position in h is qm 1 @x th—t
entryD the m   I entries following it —re —r˜itr—ryD the other entries —re uniquely
determinedAF ren™e the tot—l num˜er of di—gon—l entries in elements of h th—t
—re equ—l to  is pqm 1F „he expe™ted num˜er of 9s in — r—ndom mem˜er of h is
therefore pqm 1 a pF
£
jhj
q
MINIMAL NONABELIAN GROUPS AND TRIANGULARIZATION
9
PFQF Summary. rere we summ—rize some of the results proven in se™tions PFI —nd
PFP —˜out properties th—t hold for nilpotent —s well —s nonnilpotent niteD minim—l
non—˜eli—n groupsF
Theorem 2.3.1. Let G be a nite, minimal nonabelian group. Then there exist
a; b P G, positive integers i; j, and primes p; q (not necessarily distinct), such that
ord@aA a qi, ord@bA a pj, G a ha; bi, p; q are the only primes dividing jGj, and
i a I whenever p Ta q. Furthermore
@IA Every element in the commutator group of G is a commutator.
@PA The group G is nilpotent if and only if p a q.
@QA If f is any @p; qA-polynomial (other then x C I, if p a q a P), then all
nonscalar irreducible representations of G are of size p and are given by
a U3 A, b U3  p, where qi a I a pj, and A is a nonscalar element of
h@p; qY fA.
£
Theorem 2.3.2. If q  wp@CA is an irreducible, nite, minimal nonabelian,
matrix group then q is a @p; q; jA-matrix group.
£
Corollary 2.3.3. Let p; q be primes, A a di—g @1; : : : ; pA with qr a I, ; P C
such that qi a I a pj, and q a h A;  pi. If either @p; qA Ta @P; PA, or @ ; A a
@I; IA, then
@IA Minimal nonabelian subgroups of q are all similar.
@PA If p a q then there is a unique minimal nonabelian subgroup of q.
£
QF Applications
‡e st—rt ˜y re™ording the following useful f—™t whi™h e—sily follows from the
stru™ture theory of minim—l non—˜eli—n groups we h—ve developedF
Proposition 3.0.1. (cf. ‘Q“) If q is a nite, nonabelian matrix group, then the
p
spectral radius of some ring commutator AB   BA, A; B P q is at least Q. We
can further assume that this ring commutator commutes with every element of the
derived subgroup ‘q; q“.
Proof. ‡ithout loss of gener—lity we —ssume th—t q is — minim—l non—˜eli—n groupF
xote th—t AB   BA a AB@I   A 1B 1ABAF ren™e for every D P ‘q; q“ some
ring ™ommut—tor h—s the spe™tr—l r—dius equ—l to the spe™tr—l r—dius of I   DF £
‚e™—ll th—t if G is — groupD then g P G is ™—lled — PEelement if g2r a I for some
integer rF
10
MITJA MASTNAK AND HEYDAR RADJAVI
Lemma 3.0.2. Let q be a nite, minimal nonabelian matrix group. If the spectrum
of every group commutator ‘X; Y “ a XY X 1Y  1 contains at most two elements,
then either q contains a noncentral P-element, or q is nilpotent.
Proof. st is su™ient to prove the st—tement when q a q@p; qY Y fA is — @p; q; jAE
m—trix group with p Ta qF ‡e use the f—™t th—t every element of the ™ommut—tor
su˜group is — ™ommut—torF ƒuppose if possi˜leD th—t p Ta P Ta qF vet f@xA a
xm C am 1xm 1 C : : : C a1x C a0F sf m ! QD then rst three di—gon—l entries of
every element of h@p; qY fA a ‘q; q“ —re —r˜itr—ry qEth roots of unityY ™ontr—di™ting
n

o
the —ssumptionF sf m a ID then h@p; qY fA a di—g ; ; : : : ; p 1
—nd we
—g—in h—ve — ™ommut—tor with —t le—st Q distin™t eigenv—luesF essume now th—t
m a P —nd f@xA a x2 C a1x C a0F ‡ithout loss of gener—lity we m—y —ssume
th—t neither a0D nor a1 is divisi˜le ˜y q @‚em—rk IFIFPAF sf  is — primitive pEth
root of unityD then di—g @I; ;  a1; : : :A ; di—g @; I;  a0; : : :A P hF sf either a0 or a1
is dierent from  I P FqD then we get — ™ontr—di™tionF sf a0 a a1 a  ID then
di—g @;  1; I; : : :A P h@p; qY fAF eg—in — ™ontr—di™tionF
£
Lemma 3.0.3. If q is an irreducible, nite, minimal nonabelian, matrix group and
the spectrum of every ring commutator AB BA, A; B P q is an R-collinear subset
of C, then the spectrum of every group commutator has at most two elements.
Proof. sf A a diag@1; : : : ; pA P q a q@p; qY Y fAD then A@  pA   @  pAA h—s
eigenv—lues @i   i+1AF sf —ll these v—lues —re RE™olline—r this me—ns th—t —t most
two of i9s ™—n ˜e distin™tF
£
Corollary 3.0.4. Let q be a compact group of matrices. Assume that the spectrum
of every ring commutator AB   BA, A; B P q is an R-collinear subset of C. If q
contains no noncentral P-elements, then q is abelian.
Proof. essume th—t q ™ont—ins no non™entr—l PEelements —nd suppose th—t q is not
—˜eli—nF ƒin™e — ™omp—™t group is —˜eli—n if —nd only if every nite su˜group is su™h
‘Q“D we m—yD without —ny loss of gener—lityD —ssume th—t q is — minim—l non—˜eli—n
groupF sf q is not nilpotentD then we —re done ˜y vemm— QFHFQ —nd goroll—ry
QFHFRF essume now th—t q is nilpotentF „hen q is — pEgroup for some prime p Ta PF
flo™k di—gon—lize qD note th—t one of the nons™—l—r irredu™i˜le ˜lo™ks is simil—r
to — @p; p; jAEm—trix group q@p; pY Y fA a hA; BiD where A a di—g @I; ; : : : ; p 1AD
B a  pD —nd o˜serve th—t the spe™trum of AB   BA is not RE™olline—rF
£
sf ƒ is — semigroupD then R+ƒ a frSj r > H; S P ƒg denotes its positively hoE
mogeneous ™losureF
Theorem 3.0.5. Let ƒ a R+ƒ be an irreducible semigroup of matrices and
let L be the convex hull of the spectra of all ring commutators in ƒ, i.e., L a
co f P @ST   T SAj S; T P ƒg. If L Ta C, then L a R, or L a iR. Furthermore,
if ƒnfHg is a group and L a R, then ƒ contains a noncentral involution.
MINIMAL NONABELIAN GROUPS AND TRIANGULARIZATION
11
Proof. y˜serve th—t if L Ta CD then L a RD for some nonzero P CF vet k
˜e the minim—l positive r—nk in ƒ —nd let E ˜e —n idempotent of r—nk kF sf
k > I then ™onsider EƒEjRange(E)nfHg a R+qD where q is — ™omp—™t groupF
‡ithout loss of gener—lity we —ssume th—t q is — su˜group of unit—ry m—tri™esF fy


goroll—ry QFHFR q ™ont—ins — non™entr—l PEelement U a I1
H
H  I F vet V P q
2
˜e —n element whi™h does not ™ommute with U —nd note th—t H a ‘UV £; V “r a
@U   V UV £A is — nonzero hermiti—n m—trix @indeedD sin™e U£ a 2UD we h—ve
H£ a @U£ C V U£V £A a 2@U C V UV £A a HAF ren™e L a R —nd sin™e 2I P ƒ
we —lso h—ve 2L a LY whi™h in turn implies th—t 2 P RF
xow —ssume th—t k a IF „hen it is su™ient to ™he™k the ™l—im for hA; Bi




‘T“D where A a a H
H I
I H —nd B a H b D with ab Ta IF xote th—t ‘A; B“r a


AB  BA a I  a
b  I F „he ring ™ommut—tor ‘A; B“r is not nilpotent sin™e ab Ta IF
„hus we h—ve a; b P RD sin™e a‘A; B“r a ‘A2; B“r —nd b‘A; B“r a ‘A; B2“F ren™e
L a R @if ab < IA or L a iR @if ab > IAF
£
Remark 3.0.6. The proof above also shows that both cases L a R and L a iR
are possible.
„he following numeri™—l inv—ri—nt @fA of — polynomi—l f will ˜e useful in studyE
ing @preAtri—ngul—rizing ™onditions on semigroups of oper—torsF
De nition 3.0.7. For a function f X C 3 C we de ne
n

o
@fA a inf m—x jf@Aj p prime :
p=1
Proposition 3.0.8. If f X C 3 C is a continuous function, then the following are
equivalent.
@IA For every prime p there is a  P C such that p a I and f@A Ta H.
@PA @fA > H.
Proof. vet M a m—x jf@Aj —nd let 0 P S1 ˜e su™h th—t f@0A a MF vet  > H ˜e
jj=1
su™h th—t for j   0j >  we h—ve jf@Aj > M=P —nd let p0 ˜e — prime su™h th—t
for —ll primes p ! p0 there is — pEth root of I in the Eneigh˜orhood of 0F „hen
&

'
M

@fA ! min
; m—x f@A
> H
P p=1
 p prime; p < p0
£
Remark 3.0.9. If f X C 3 C is an analytic function, then the condition (1) from
the proposition above says that f is not divisible by xp   I for any prime p.
12
MITJA MASTNAK AND HEYDAR RADJAVI
‡hen f is — polynomi—l then the following result provides s su™ient ™onditionD
p
independent of the degree of fD for Q @fAF
Proposition 3.0.10. Let f be a monic polynomial such that f@IA a H. If no roots
of f lie in the region @
p
A
Q
I
D a z P Cj Re@zA <
  ; H < jzj < P
Q
P
p
then @fA ! Q.
Proof. vet f@xA a @x IA@x 1A : : : @x nA —nd let p ˜e — primeF vet  ˜e the pEth
root of unity ™losest to  I with nonneg—tive im—gin—ry p—rtD iFeFD  a e(p 1)i —nd
note th—t j@ IA@ IAj ! Q —nd th—t for —ny  TP D we h—ve j@ A@ Aj ! IF „he
p
l—tter is ™le—r if either  a H or jj ! PF xow —ssume  is su™h th—t 3   1 Re@AF
p 3
2
xote th—t it is su™ient to prove the st—tement for Re@A a 3   1F ix—mine
p
3
2
the tri—ngle with verti™es ;  —nd F vet h a 3   1   Re@A —nd s a Im@A a
3
2
 Im@A ! HF „he —re— A of the tri—ngle in question is hsF vet a a j   j —nd
b a j   jF sf A ! 1D then we —re done —s ab ! PAF xote th—t if p a QD then
2
PA a I —nd th—t if p a SD then PA % I:HR > IF xote —lso th—t if p ! UD then h > sF
sn this ™—se let ˜e the —ngle —t  —nd let H ˜e the —ngle —t H Xa Re@A of the
p
tri—ngle @; ; HA —nd note th—t < H < F hene d a h2 C s2 —nd o˜serve th—t
2
in this ™—se ab a d2 sin( ) ! d2F pin—lly if p ! UD then d2 ! IF sndeedD if p a UD then
sin( H)
d2 % I:IR > IF sf p ! IID then —lre—dy h > I @if p a IID then h % I:HQ > IAF
£
Remark 3.0.11. With some care the region D in the theorem above could be
shrunk to
¨


©
DH a z P CjV prime p; @z   e(p 1)iA@z   e(p+1)iA ! I :
p
The quantity 3   1 % H:HUU is fairly close to H.
3
2
De nition 3.0.12. If g P Chx; yi is a polynomial in two noncommuting variables,
then de ne g1; g2 X C 3 C by g1@tA a g@t; IA, g2@tA a g@I; tA, 1@gA a @g1A,
2@gA a @g2A and @gA a m—x f1@gA; 2@gAg.
Theorem 3.0.13. Let g be a homogeneous polynomial in two noncommuting vari-
ables of joint degree r. If H k < @gA, then every nite group q of matrices
satisfying
@g@xy; yxAA k@xAr@yAr;
for all x; y P q is abelian.
Proof. essume with no loss of gener—lity th—t @gA a 1@gA ƒuppose th—t q is
— minim—l non—˜eli—n group s—tisfying @g@xy; yxAA k@xAr@yAr a kF ren™e
@‘x; y“; IA kD for —ll x; y P GF fy the stru™ture theoremD every element of the
MINIMAL NONABELIAN GROUPS AND TRIANGULARIZATION
13
™ommut—tor su˜group is — ™ommut—tor —nd for some prime pD every pEth root of
unity is in the spe™trum of some ™ommut—torF ren™e we h—ve
1@gA m—x jg@I; tAj k < @gA:
p=1
„his is ™le—rly — ™ontr—di™tionF
£
‡e ™—n extend „heorem QFHFIQ to the gener—l semigroup settingF ‚e™—ll th—t —
property € th—t — semigroup ƒ of ™omplex m—tri™es m—y possess is ™—lled pretriE
—ngul—rizing ‘I“ if the following holdsX
@IA € is simil—rity inv—ri—ntF
@PA € p—sses to su˜semigroupsD homogenized ™losures —nd semisimpli™—tionsF
@QA sf ƒ ¨ H h—s property €D then so does ƒF
@RA „ot—lly redu™i˜le semigroups with € h—ve no nonEzero nilpotentsF
@SA pinite groups with € —re —˜eli—nF
st should ˜e noted th—t re—son—˜le m—trix semigroup properties s—tisfy the ™ondiE
tions @IAE@QA —nd th—t it is usu—lly su™ient to ™he™k @RA for irredu™i˜le semigroupsF
sn ‘I“ it w—s proven th—t groups s—tisfying — pretri—ngul—rizing property —re —lw—ys
redu™i˜le —nd h—ve tri—ngul—riz—˜le ™ommut—tor su˜groupsF st w—s —lso shown th—t
if r is the minim—l nonzero r—nk in — semigroup ƒ a R+ƒ s—tisfying — pretri—nguE
l—rizing propertyD then ƒ h—s — ™h—in of inv—ri—nt su˜sp—™es of length —t le—st rF st
is impli™it in the proof of ‘ID „heorem SFI“ th—t the l—st st—tement ™—n ˜e extended
to semigroups of ™omp—™t oper—tors on — f—n—™h sp—™eF wore pre™iselyD if ˆ is —
f—n—™h sp—™e —nd ƒ a R+ƒ is — semigroup of ™omp—™t oper—tors on ˆ s—tisfying —
property €D whi™h is pretri—ngul—rizing for m—trix semigroupsD then ƒ h—s — ™h—in
of ™losed inv—ri—nt su˜sp—™es of length —t le—st rY where r is the minim—l nonzero
r—nk @possi˜ly inniteA in ƒF
Theorem 3.0.14. Let g be a homogeneous polynomial in two noncommuting vari-
ables of joint degree r such that either g@t; HA Ta H or g@H; tA Ta H. If H k < @gA,
then the property
@g@xy; yxAA k@xAr@yAr; for all x; y P ƒ
for matrix semigroups ƒ is a pretriangularizing property.
Proof. gonditions @IAD@PAD —nd @QA —re ™le—rly s—tisedF gondition @SA is the the
™ontent of „heorem QFHFIQF xote th—t it is su™ient to ™he™k ™ondition @RA for
irredu™i˜le semigroups ƒF essumeD if possi˜le th—t ƒ is —n irredu™i˜le semigroup
s—tisfying @g@xy; yxAA k@xAr@yArD for —ll x; y P ƒ —nd th—t ƒ ™ont—ins —
nonzero nilpotent NF ‡ithout loss of gener—lity —ssume th—t ƒ a R+ƒD g@t; HA T HD
—nd th—t N2 a HF xote th—t for every X P ƒD the m—trix g@X; NA is nilpotentF
‡rite




N a H M
X1;1 X1;2
H H ; X a X2;1 X2;2
14
MITJA MASTNAK AND HEYDAR RADJAVI
—nd note th—t the m—tri™es g@MX2;1; HA —nd g@H; X2;1MA —re nilpotentF ren™e
gn@MX2;1; HA a H a gn@H; X2;1MAF „hus the union of spe™tr— of —ll MX2;1 is —
nite setY —nd hen™e these spe™tr— —re fHg due to homogeneity of ƒF „herefore
f X wn 3 CD f@XA a trMX2;1 denes — nonzero fun™tion—l whi™h is H on ƒF „his
™ontr—di™ts the irredu™i˜ility of ƒF
£
Remark 3.0.15. The case g@t; sA a t   s was already investigated in ‘Q“.
prom the theorem —˜ove we o˜t—in the following ™oroll—ryF
Theorem 3.0.16. Let g be a homogeneous polynomial in two noncommuting vari-
ables of joint degree r, such that either g@t; HA Ta H or g@H; tA Ta H. Let ƒ be any
semigroup of compact operators on a Banach space such that
@IA
@g@AB; BAAA k@AAr@BAr;
for all A; B P ƒ and some xed k, with H k < @gA. Then either ƒ is reducible
or it contains operators of rank I.
Proof. „his follows from the f—™t th—t @IA is — pretri—ngul—rizing property for m—E
trix semigroups ™om˜ined with the dis™ussion following „heorem QFHFIQF
£
sf g is rigid in the sense of ‘T“D then t—king a H we o˜t—in the tri—ngul—rizing
™ondition of „heorem RFP of ‘T“F
Acknowledgement
‡e would like to th—nk the referee for dire™ting us to the origin—l p—per ‘R“F
References
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[2] J. Bernik, R. Guralnick, M. Mastnak, Reduction theorems for groups of matrices, Linear
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(2005), 309.-320.
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AMS, Vol. 4, No. 4 (1903), pp. 398-404
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197{220.
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