Permutation-like Matrix Groups with a Maximal Cycle of Power of Odd Prime Length

0.15 MB·English

by  Guodong Deng

Permutation-like Matrix Groups with a Maximal
5
1 Cycle of Power of Odd Prime Length
0
2
Guodong Deng, Yun Fan
y
a
M School of Mathematics and Statistics
Central China Normal University,Wuhan 430079, China

9

]
R
G
Abstract
.
h Ifeveryelementofamatrixgroupissimilar toapermutationmatrix,
t then it is called a permutation-like matrix group. References [3] and [4]
a
m showedthat,ifapermutation-likematrixgroupcontainsamaximalcycle
of length equal to a prime or a square of a prime and the maximal cycle
[
generates a normal subgroup, then it is similar to a permutation matrix

3 group. In this paper, we prove that if a permutation-like matrix group
v contains a maximal cycle of length equal to any power of any odd prime
6 and the maximal cycle generates a normal subgroup, thenit is similar to
5 a permutation matrix group.
3
1 Keywords: permutation-likematrixgroup,permutationmatrixgroup,
0 maximal cycle.
.
1 MSC2010: 15A18, 15A30, 20H20.
0
5
1 1 Introduction
:
v
i
X Let C be the complex filed. Let GL (C) be the complex generallinear groupof
d
r dimensiond, i.e. the multiplicative groupconsisting of invertible complex d×d
a matrices. Any subgroup G ≤ GL (C) is called a matrix group of dimension d.
d
For a matrix group G, if there exists a P ∈ GL (C) such that P−1AP for all
d
A∈G are permutation matrices, then G is said to be similar (or conjugate) to
a permutation matrix group, orsaidto be apermutation matrix groupforshort.
If for every element A of G there exists an S ∈ GL (C) such that S−1AS is a
d
permutation matrix, then G is called a permutation-like matrix group.
Cigler [2, 3] showed that a permutation-like matrix group is not a permu-
tation matrix group in general. A d×d matrix is called a maximal cycle if it
is similar to a permutation matrix corresponding to the cycle permutation of
length d. Cigler conjectured that:
Email address: [email protected](YunFan).
1
Conjecture: If a permutation-likematrix groupcontains a maximalcycle, then
it is a permutation matrix group.
Cigler [2, 3] proved that, if a permutation-like matrix group G of dimension
prime p contains a maximal cycle which generates a normal cyclic subgroup,
thenG is a permutationmatrixgroup. In [4], we further provedthat this result
still holds if we replace the dimension prime p by dimension p2.
In this paper we prove:
Theorem 1.1. LetG beapermutation-likematrix groupofdimension pn where
p is any odd prime and n is any positive integer. If G contains a maximal cycle
whichgeneratesanormalcyclic subgroup,thenG isapermutationmatrixgroup.
Necessary preparations for the proof of the theorem are made in Section 2.
For fundamentals of the group theory, please refer to [1, 5]. In Section 3 we
treataspecialcaseofthetheoremwhereG isap-group,i.e. itsorderisapower
of p. The proof of Theorem 1.1 is presented in Section 4.
2 Preparation
We begin with few preliminaries on relations between a cyclic group hCi of
orderdgeneratedbyC andtheresidueringZ oftheintegerringZmodulothe
d
positive integer d. We have a bijection which maps k ∈Z to Ck ∈hCi. By Z∗
d d
we denote the multiplicative group consisting of the reduced residue classes in
Z . Obviously,hCki=hCiifandonlyifk ∈Z∗;atthatcasewesaythatCk isa
d d
generatorofthecyclicgrouphCi. AnyautomorphismαofhCiiscorresponding
toexactlyoner ∈Z∗ suchthatα(Ck)=Crk foranyCk ∈hCi,Byµ wedenote
d r
thepermutationofZ mappingk∈Z toµ (k)=rk. Theautomorphismgroup
d d r
of hCi is isomorphic to Z∗ by mapping α to µ as above. If G is a finite group
d r
whichcontainshCiasanormalsubgroup,thenG ishomomorphictoasubgroup
of Z∗ with kernel consisting of the elements which centralize hCi.
d
For an element A of a group G, we denote the order of A by ord (A), or
G
ord(A) for short if the group is known from context. Let p be a prime. A finite
group G is called a p-group if its order |G| is a power of p. The next lemma
aboutacyclicp-groupisobvious. Asmentionedabove,itisalsoalemmaabout
Z . Note that |Z∗ |=pn−1(p−1).
pn pn
Lemma 2.1. Let G =hCi be a cyclic group and |G|=pn.
(i) For 0≤a <n denote Gpa = {Cpat | 0≤t <pn−a}. Then Gpa =hCpai
is a cyclic subgroup of G of order pn−a generated by Cpa.
(ii) The map G →Gpa, Ck 7→Cpak, is an epimorphism with kernel Gpn−a.
In particular, for any generator G of the cyclic group Gpa there are exactly pa
generators of G which are mapped to G.
(iii) Ifpis an odd prime andαis an automorphism ofG, thenord(α)=spa
with s | (p−1) and 0 ≤ a < n, and there are integers u,v coprime to p such
that ordZ∗ (u)=s (hence ordZ∗(u)=s) and α(C)=Cu+vpn−a.
pn p
2
For a subgroup H of a group G, if any element which centralizes H is con-
tained in H, then H is said to be self-centralized.
Lemma 2.2. Assume that G is a finite group containing a self-centralized nor-
mal cyclic p-subgroup hCi generated by an element C, where p is an odd prime.
Then G =hA,Ci generated by two elements, and one of the following two holds.
(i) If G is a p-group, then there is an A′ ∈ G such that G = hA′,Ci and
hA′i∩hCi=1.
(ii) If G is not a p-group, then hAi∩hCi=1.
Proof. We sketch a proof, some of the arguments will be quoted later.
Assume that |hCi| = pn. Since hCi is self-centralized, the quotient group
G/hCi ∼= hAi/hAi∩hCi is isomorphic to a subgroup of the multiplicative Z∗ .
pn
The groupZ∗ is a cyclicgroupoforderpn−1(p−1). Thus, |G/hCi|=spa with
pn
0≤a<n and s|(p−1), and there is an r ∈Z∗ such that
pn
A−1CA=Cr and ordZ∗ (r)=|G/hCi|=spa. (2.1)
pn
For any integer k and positive integer j, it is a direct computation that
(ACk)j =(ACk)···(ACk)(ACk)=AjCk(rj−1+···+r+1). (2.2)
(i). Assume that |G/hCi| = pa, 0 < a < n. By Lemma 2.1(iii), we assume
that r=1+vpn−a with p∤v. Denote hCiA ={Ct |A−1CtA=Ct}. Note that
A−1CtA = Ctr. So A−1CtA = Ct if and only if t(r−1) ≡ 0 (mod pn). Thus
hCiA = hCpai. Since G/hCi ∼= hAi/hAi∩hCi, Apa ∈ hCi hence Apa ∈ hCiA =
hCpai. So, we can find an integer k such that ApaCkpa =1. Note that
rpa −1 (1+vpn−a)pa −1
rpa−1+···+r+1= = ≡pa (mod pn). (2.3)
r−1 vpn−a
Let A′ =ACk. By Eqns (2.2) and (2.3), we obtain
A′pa =ApaCk(rpa−1+···+r+1) =ApaCkpa =1.
Thus, G =hA′,Ci and hA′i∩hCi=1.
(ii). Assume that|G/hCi|=spa, 0≤a<n ands|(p−1). Since G is nota
p-group, we have s>1. By Lemma 2.1 (iii), we can assume that ordZ∗(r) =s.
p
Then p∤(r−1) (otherwise s=1). As we have seen, A−1CtA=Ct if and only
if t(r−1) ≡ 0 (mod pn). At the present case, t(r −1) ≡ 0 (mod pn) if and
only if t ≡ 0 (mod pn). Thus hCiA = 1. As argued above, Aspa ∈ hCiA. So
Aspa =1. That is, hAi∩hCi=1.
We turn to preliminaries on matrices.
B
1
A diagonal blocked matrix  ...  is denoted by B1 ⊕···⊕Bm
B
 m
for short. The identity matrix of dimension d isdenoted by Id×d, or I for short
if the dimension is known from context.
WedenotethecharacteristicpolynomialofacomplexmatrixAbychar (x).
A
3
Lemma 2.3 ([4, Lemma 2.1]). The following two are equivalent to each other:
(i) A is similar to a permutation matrix;
(ii) A is diagonalizable and charA(x)= i(xℓi −1).
If it is the case, then each factor xℓi −1 oQf charA(x) corresponds to exactly
one ℓ -cycle of the cycle decomposition of the permutation of the permutation
i
matrix.
By Φ (x) we denote the cyclotomic polynomial of degree n, i.e. Φ (x) =
n n
(x−ω)withω runningovertheprimitiven-throotsofunity. Sincexn−1=
ω
Φ (x), the following is an immediate consequence of the above lemma.
Qk|n k
CQorollary. Let A be a matrix similar to a permutation matrix, and m,n be
positiveintegers. IfΦ (x)m char (x),thenΦ (x)m char (x)foranyk|n.
n A k A
The next lemma is a com(cid:12)bination of [4, Eqns (2.(cid:12)1), (2.2), (2.3) and (2.4)].
(cid:12) (cid:12)
Lemma 2.4. Let C ∈GL (C) be a maximal cycle of dimension d, and λ be a
d
primitive d-th root of unity. Then the following hold.
(i) {λj | j ∈ Z } = {λ0 = 1,λ,··· ,λd−1} is the spectrum (i.e. the set of
d
eigenvalues) of C.
(ii) The eigen-subspace of every eigenvalue λj of C, denoted by E(λj), has
dimension 1, and Cd = d−1E(λj).
j=0
(iii) Ife ∈E(λj)forj =0,1,··· ,d−1areallnonzero,then(e ,e ,··· ,e )
j L 0 1 d−1
is a basis of Cd and Ce =λje , j =0,1,··· ,d−1.
j j
(iv) Let f =α e +α e +···+α e where e ,··· ,e are as above
0 0 1 1 d−1 d−1 0 d−1
and all α ∈ C. Then (f, Cf, ··· , Cd−1f) is a basis of Cd if and only if
j
α 6=0 for all j =0,1,··· ,d−1. If it is the case, with respect to that basis, C
j
is a cycle permutation matrix of dimension d.
Lemma 2.5 ([4, Lemma 2.2]). Let C and λ be as above in Lemma 2.4. Let
A∈GL (C) such that A−1CA=Cr for an r ∈Z∗. Further assume that Z is
d d d
partitioned into µ -orbits as follows:
r
Γ ={0}, Γ ={j ,rj ,··· ,rd1−1j }, ··· , Γ ={j ,rj ,··· ,rdm−1j }
0 1 1 1 1 m m m m
i.e. rdkjk ≡ jk (mod d) but rdk−1jk 6≡ jk (mod d). For k = 0,··· ,m, take
nonzero ejk ∈ E(λjk), set Ek = {ejk,Aejk,··· ,Adk−1ejk} and E = mk=0Ek.
Then the following hold.
S
(i) A·E(λj)=E(λrj), j =0,··· ,d−1,whereA·E(λj)={Av |v ∈E(λj)}.
j j
(ii) Ek is a basis of Vk := hdj=−01E(λrhjk), and A restricted to Vk has the
matrix
L0 ··· 0 ω
k
1 ... ... 0 
A| = , (2.4)
Ek ... ... ... ... 
 
0 ··· 1 0 
 dk×dk
 
4
where ω is an (ord(A)/d )-th root of unity,
k k
(iii) Cd =V ⊕···⊕V , the union E =E ∪···∪E is a basis of Cd and,
1 m 1 m
with respect to the basis E, the matrix of A is
A| =A| ⊕···⊕A| .
E E1 Em
Proposition 2.6 ([4, Proposition2.3]). Let the notations be as in Lemma 2.5.
If the matrix A| is a permutation matrix (i.e. all ω = 1), then the matrix
E k
group hA,Ci generated by A and C is a permutation matrix group.
Note that “the matrix A| is a permutation matrix” means that mapping
E
e∈E to Ae is a permutation of the set E.
Lemma 2.7 ([3, Proposition4.2]). If G is a permutation-like matrix group and
C ∈G is a maximal cycle, then hCi is self-centralized in G.
We sketchthe prooffor convenience. For A∈G whichcentralizesC, wecan
assume that C is a permutation matrix of dimension d corresponding to the d-
cyclepermutationandA= d−1α Ci. SinceACd−kissimilartoapermutation
i=0 i
matrix, its trace dα is a non-negativeinteger. Hence α for k =0,1,··· ,d−1
k k
P
are non-negative rationals. By Lemma 2.3, all the eigenvalues of A are roots
of unity, in particular, d−1α = 1 = d−1α λi . If there are at least
i=0 i i=0 i
two of the coefficients non-zero, say α 6= 0 6= α for 0 ≤ k 6= h < d, then
α λk+α λh <α +αPbecause λk 6=kλh(cid:12)(cid:12);Phenceh (cid:12)(cid:12)
k h k h
(cid:12) (cid:12)
(cid:12) (cid:12)d−1 d−1
1= α λi ≤ α λk+α λh + α λi < α =1,
i k h i i
(cid:12)(cid:12)(cid:12)Xi=0 (cid:12)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) i6=Xk,h(cid:12)(cid:12) (cid:12)(cid:12) Xi=1
which is impossible. Thus there is exactly one of the coefficients, say α , which
k
is non-zero; then α =1 and A=Ck.
k
3 The case of p-groups
In the following p is always an odd prime and n is a positive integer. In this
section we consider a special case of Theorem 1.1, where the permutation-like
matrix group is a p-group. This is a key step for the proof of the theorem.
Let C ∈ GL (C) be a maximal cycle. We have a disjoint union Z =
pn pn
pZ Z∗ , where
pn pn
S pZ ={pk (mod pn)|k∈Z }={0,p,··· ,p(pn−1−1)}.
pn pn
ByLemma2.4,wehavetwosubspacesofCpn,denotedbyVpandV∗,asfollows:
Vp = E(λj)= Ce and V∗ = E(λj)= Ce (3.1)
j∈MpZpn eM∈Ep j∈MZ∗pn eM∈E∗
5
where Ep and E∗ denote the basis of Vp and V∗ respectively as in Lemma 2.5.
Then Cpn =Vp⊕V∗.
For any A ∈ GL (C) which normalizes hCi, there is exactly one r ∈ Z∗
pn pn
such that A−1CA = Cr. Both pZ and Z∗ are µ -invariant. So, for any
pn pn r
integerk with0<k<pn bothVp andV∗ areACk-invariantsubspacesofCpn.
ByACk|Vp andACk|V∗ wedenotethelineartransformationsofACk restricted
to Vp and V∗ respectively. Correspondingly, ACk|Ep and ACk|E∗ are matrices
of ACk|Vp and ACk|V∗ respectively.
Lemma 3.1. Let A,C ∈ GL (C). Assume that C is a maximal cycle and
pn
A−1CA=Cr for an r ∈Z∗pn with ordZ∗pn(r)=pa where 0≤a<n. Let Vp and
V∗ be as in Eqn (3.1). If Apa =I, then for
Φ (x)pνp(k), 0≤ν (k)<n−a;
pn−νp(k) p
char (x)=
(ACk)|V∗ (xpa −1)pn−a−1(p−1), ν (k)≥n−a, or k =0;
 p
where νp(k) denotes thep-adic valuation of k, i.e. pνp(k) is the largest power
of p which divides k.
Proof. Assume that Z∗ is partitioned into µ -orbits Γ , ···, Γ . Every orbit
pn r 1 h
Γ has length pa, and the number h=pn−a−1(p−1). We can assume that
i
Γ ={j ,rj ,··· ,rpa−1j }, ··· , Γ ={j ,rj ,··· ,rpa−1j }.
1 1 1 1 h h h h
For the basis E∗ of V∗, by Lemma 2.5 we have E∗ =E∗∪···∪E∗ with
1 h
E∗ = e , Ae , ··· , Arpa−1e , i=1,··· ,h. (3.2)
i ji ji ji
Denote the restrict(cid:8)ed matrices to Ei∗ by Ai (cid:9)= A|Ei∗ and Ci = C|Ei∗ for i =
1,··· ,h. Since Apa =I, we have
i
0 ··· 0 1 λji
1 ... ... 0 λrji
Ai =0... ·.·.·. .1.. 0... , Ci = ... λrpa−1ji . (3.3)
 pa×pa  pa×pa
   
Thus ACk|E∗ = hi=1AiCik, and
charAiLCik(x)=xpa −λPj∈Γijk =xpa −λjik(1+r+···+rpa−1).
The conclusion is obviously true if k = 0. So we further assume that k 6= 0.
By Lemma 2.1 (iii), we can take an integer v which is coprime to p such that
r = 1+vpn−a. Then it is easy to check that ν (1+r+···+rpa−1) = a, see
p
6
Eqn (2.3). So we can write 1+r+···+rpa−1 = a′pa and k = k′pνp(k) with
p∤a′ and p∤k′. Then
char (x)=xpa −λa′k′jipνp(k)+a, i=1,··· ,h.
AiCik
By Lemma 2.1 (ii), the collection of λa′k′jipνp(k)+a for i = 1,··· ,h is just the
collection of all primitive pn−a−νp(k)-th roots of unity, each of which appears
with multiplicity
h
=pνp(k).
pn−a−νp(k)−1(p−1)
Ifn−a−ν (k)>0,thenthe collectionofrootsofchar (x)isjustthe col-
p ACk|V∗
lection ofall primitive pn−νp(k)-th roots of unity, each ofwhich has multiplicity
pνp(k); hence
char (x)=Φ (x)pνp(k).
ACk|V∗ pn−νp(k)
Otherwise, n−a−νp(k)≤0, i.e. λa′k′jipνp(k)+a =1, hence
char (x)=(xpa −1)h =(xpa −1)pn−a−1(p−1). (cid:3)
ACk|V∗
Corollary 3.2. Let notation be as in Lemma 3.1. If the matrix group hA,Ci is
permutation-like and Apa = I, then A|Vp,C|Vp is a permutation-like matrix
group of dimension pn−1.
(cid:10) (cid:11)
Proof. Let ℓ,k be any non-zero integers. Let A′ = Aℓ, a′ = a −ν (ℓ) and
p
r′ =rℓ. Then A′−1CA′ =Cr′ and ord(A′)=ordZ∗ (r′)=pa′. by Lemma 3.1,
pn
Φ (x)pνp(k), ν (k)<n−a′;
pn−νp(k) p
char (x)=
(AℓCk)|V∗ (xpa′ −1)pn−a′−1(p−1), νp(k)≥n−a′.
Since AℓCk = A′Ck is similar to a permutation matrix, by Lemma 2.3 and its
corollary,
(xpn−νp(k)−1 −1)pνp(k), ν (k)<n−a′;
char (x)= p
AℓCk|Vp  (xpi −1)ji, ν (k)≥n−a′.
 i p
Q
By Lemma 2.3 again,the matrix group A| ,C| is a permutation-like ma-
Vp Vp
trix group of dimension pn−1.
(cid:10) (cid:11)
Lemma 3.3. Let A,C ∈ GLpn(C). Assume that C is a maximal cycle and
A−1CA = Cr for an r ∈ Z∗pn with ordZ∗pn(r) = p. If hA,Ci is a permutation-
like matrix group and Ap =I, then A| =I.
Vp
7
Proof. Note that hA,Ci={AℓCk |0≤ℓ<p, 0≤k <pn}. For any 0<ℓ<p
and 0<k <pn, by Lemma 3.1 (note that a=1 at present case),
Φ (x)pνp(k), ν (k)<n−1;
char (x)= pn−νp(k) p
AℓCk|V∗ ((xp−1)pn−2(p−1), νp(k)≥n−1.
Since AℓCk is similar to a permutation matrix, by Lemma 2.3,
(xpn−νp(k)−1 −1)pνp(k), ν (k)<n−1;
char (x)= p (3.4)
AℓCk|Vp (x−1)pn−1−pj(xp−1)j, ν (k)≥n−1.
 p
Thus A|Ep,C|Ep is a permutation-like matrix group of dimension pn−1.
Obviously, C| = λj is a maximal cycle of dimension pn−1. Since
(cid:10) (cid:11)Ep j∈pZpn
r = 1+vpn−1 for some integer v coprime to p (see Lemma 2.1(iii)), for any
L
pt∈ pZ we have µ (pt)≡ pt (mod pn). So, A| is a diagonal matrix, hence
pn r Ep
A|Ep commutes with C|Ep. By Lemma 2.7, A|Ep ∈ C|Ep . But (A|Ep)p = I
and ord(C|Ep) = pn−1. Thus A|Ep ∈ CEpnp−2 , and we(cid:10)can(cid:11)assume that A|Ep =
(C| )bpn−2 with 0≤b<p. Suppose that b>0, then ν (−bpn−2)=n−2, and
Ep (cid:10) (cid:11) p
by Eqn (3.4),
charIpn−1×pn−1(x)=char(A|Ep)(C|Ep)−bpn−2(x)=(xp−1)pn−2.
However, char (x)=(x−1)pn−1. This is a contradiction. Thus b=0
Ipn−1×pn−1
and A|Ep =Ipn−1×pn−1.
Proposition 3.4. Let A,C ∈ GL (C). Assume that C is a maximal cycle
pn
and A−1CA = Cr for an r ∈ Z∗pn with ordZ∗pn(r) = pa where 0 ≤ a < n. If
G = hA,Ci is a permutation-like matrix group and Apa = I, then A| is a
E
permutation matrix.
Proof. If n=1, then a=0 and A=I, the proposition holds trivially.
Assume n > 1. From Eqn (3.3) we have seen that A|V∗ is a permutation
matrix. By Corollary 3.2, hA| ,C| i is a permutation-like matrix group of
Vp Vp
dimension pn−1. Note that ordG(Apa−1) = ordZ∗pn(rpa−1) = p. By Lemma 3.3,
(A|Vp)pa−1 = I. And, ordZ∗ (r) = pa−1. By induction on n, A|Ep is a
pn−1
permutation matrix. Hence A|E =A|Ep ⊕A|E∗ is a permutation matrix.
Corollary 3.5. Let A,C ∈ GLpn(C). Assume that C is a maximal cycle and
A normalizes hCi. If G = hA,Ci is a permutation-like matrix p-group, then G
is a permutation matrix group.
Proof.ByLemma2.7,hCiisself-centralizedinG. ByLemma2.2(i),wehavean
A′ ∈GsuchthatG =hA′,Ci,A′−1CA′ =Cr foranr ∈Z∗pn withordZ∗pn(r)=pa
andA′pa =I. Thus,byProposition3.4andProposition2.6, G isapermutation
group.
8
4 Proof of Theorem 1.1
Let G ≤ G (C) be a permutation-like matrix group of dimension pn where p
pn
is an odd prime. Let C ∈ G be a maximal cycle such that hCi is a normal
subgroup of G. By Lemma 2.7, hCi is self-centralized in G. By Lemma 2.2 and
Eqn (2.1), G =hA,Ci, A−1CA=Cr where r ∈Z∗ with
pn
ordZ∗ (r)=|G/hCi|=spa, s|(p−1), 0≤a<n.
pn
Let E be the basis of Cpn as in Lemma 2.5.
Case 1: s=1. So G is a p-group, and Theorem 1.1 holds by Corollary 3.5.
Case 2: a = 0 and s > 1. By Lemma 2.1 (iii), we can assume that
ordZ∗(r) = s. Then for any 0 < ℓ < s, rℓ 6≡ 1 (mod p), i.e. p ∤ (rℓ −1). Thus
p
rℓk ≡k (mod pn) if and only if k ≡0 (mod pn). So every µ -orbit on Z has
r pn
length s except for the orbit {0}. By Lemma 2.2 (ii), As = I. By Lemma 2.5
(ii) and (iii),
0 ··· 0 1 0 ··· 0 1
1 ... ... 0 1 ... ... 0
A| =ω ⊕ ⊕···⊕ .
E 0 .. .. .. .. .. .. .. ..
. . . . . . . .
   
0 ··· 1 0 0 ··· 1 0
 s×s  s×s
   
And the characteristic polynomial
char (x)=(x−ω )(xs−1)(pn−1)/s.
A 0
By Lemma 2.3, ω = 1. So A| is a permutation matrix. By Proposition 2.6,
0 E
we obtain that G is a permutation matrix group.
Case 3: a > 0 and s > 1. Then G is not a p-group. By Lemma 2.2(ii)
we further have that Aspa = I. Since s and pa are coprime each other, we
have integers t,m such that st +pam = 1. Let A′ = Ast and A′′ = Apam.
Then A = Ast+pam = A′A′′, A′pa = I and A′′s = I. By Proposition 3.4, A′|
E
is a permutation matrix. From the above argument of Case 2, A′′| is also
E
a permutation matrix. Thus A| = A′| ·A′′| is a permutation matrix. By
E E E
Proposition 2.6, G is a permutation matrix group.
Acknowledgments
The authors are supported by NSFC through the grant number 11271005.
Thanks are given to the reviewers for the helpful suggestions.
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