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Can. J. Math., Vol. XXIII, No. 1, 1971, pp. 90-102

ON THE SMALLEST DEGREES OF PROJECTIVEREPRESENTATIONS OF THE GROUPS PSL(ny q)

MORTON E. HAR RIS AND CHRISTOP H HE RI NG

Introduction. In this paper, we obtain information about the minimaldegree 5 of any non-trivial projective representation of the group PSL(w, q)with n ^ 2 over an arbi trary given field K. Our main results for the groupsPSL(w, q) (Theorems 4.2, 4.3, and 4.4) state that, apart from certain excep

tional cases with small n, we have the following rather surprising situation:if q = pf (where p is a prime integer) and char K p, then 8 = n, but ifq = pf and char K ^ p, then is of a considerably higher order of magnitude,namely, is at least qn~l 1 or J (q 1) if n = 2 and q is odd. Note that forn = 2, this lower bound for

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PROJECTIVE REPRESENTATIONS OF PSL(w, q) 91

results (Lemmas 1.3 and 1.4) needed in 3, and to extend to fields of arbitrarycharacteristic, some basic results on representations of Frobenius groups over

fields of characteristic not dividing the order of the Frobenius group.Let & denote a finite group and let L denote a splitting field of characterist ic

zero of &. Let K = {Ki = {}, K2, . . . , Kk] denote the set of conjugacyclasses of S , let X = { i = 1, X2, , %k} denote a full set of representativesfor the equivalence classes of L- (absolutely) irreducible representations of ^and let ch X = {xi = 1> X2, . . . , x*} denote the corresponding characters of X(i.e., xi = t r i ) .

A very useful result of Brauer (cf. [6, Kapitel V, Satz 13.5]) is the following.

LEMMA 1.1 (Brauer). Assume that the finite group 3/ has permutation repre

sentations on: (1) the setch X, denoted by x > xA

for all A 6 s/ and x 12* denote the residue class mapping; thenthere exists a set

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92 M. E. HARRIS AND C. HERING

^-regular classes of & such that 6(

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Proof. Again it suffices to assume that F is algebraically closed. Since &acting by conjugation is transitive on 31*, Lemmas 1.1 and 1.2 and [6, Kapitel V,

Satz 20.2] imply that @ has two orbits on the ^-irreducible characters of31namely the 1 character and the other |9t| 1 characters. Thus if \F is anynon-trivial /^-irreducible character of 31, then \@ : 3(^)1 = \3l\ 1. Anappropriate modification of the proof of Lemma 1.3 yields the desired result.

Lemma 1.2 has various other applications. For example, [6, Kapitel V,Satz 16.13] which characterizes the irreducible representations of a Frobeniusgroup & over an algebraically closed field F such that char F \ \& | can begeneralized to hold for arbitrary algebraically closed fields. To prove thisgeneralization, we require the following result.

LEMMA 1.5. Let & = 31 & be a Frobenius group with Frobenius kernel 31 andcomplement &. If F is a splitting field for 31 and if % is a non-trivial irreducibleF-representation of 31, then the induced representation ^ is an absolutelyirreducible F-representation of & with kernel not containing 31 and such that

Proof. First assume that Fhas prime characteristic p. As in Lemma 1.3, ,acting by conjugation, has a faithful semi-regular permutation representationon the ^-regular classes of31. Now Lemma 1.2 applies, and we conclude that acts semi-regularly on the non-trivial irreducible ^-representations of31. Let T7*

be the algebraic closure ofFand let 36 be an (absolutely) irreducible constituenti ($9)F* such that % is equivalent to a constituent of 3E| . Now Clifford'stheorem implies that 36| is equivalent to e(Q)He$ %

H), where e is a positiveinteger.

Hence deg ^ = | | deg g ^ deg e| | deg g. Thus e = 1, ($*)F* = Xand g^ is absolutely irreducible. If Fhas characteristic zero, the analogousargument using Lemma 1.1 applies, and the lemma follows.

The generalization of [6, Kapitel V, Satz 16.13] to fields ofarbitrary characteristic mentioned above is the following.

THEOREM 1.6. Let & = 3i$ be a Frobenius group with Frobenius kernel 31 andcomplement . Let Fbe afield which is a splitting fieldforboth and31. If char Fis a prime p, letho(31) and h0(&) denote the number ofp-regular conjugate classesof 31 and respectively. If char F = 0, letho(31) andh0($>) denote the numberof conjugate classes of31 and , respectively. Then:

(1) F is a splitting field for @,(2) if $i and g2 are non-trivial irreducible F-representations of 31, then $i9

and $2^ are absolutely irreducible F-representations of&;also, %i^ andg2^are equivalent F-representations of & if andonly if there exists an H such that %\H and %2 are equivalent F-representations of31,

(3) the following list of F-representations of & comprises a full set ofrepresentatives for the set ofequivalence classes of (absolutely) irreducibleF-representations of &':

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(a) h0(&) inequivalent F-representations of & with kernel containing 9obtained from ,

(b) (Ao(9) 1)/|H inequivalent F-representations of & induced fromnon-trivial irreducible F-representations of^p.(4) if V is an irreducible F[ &]-module on which 9 acts non-trivially, then V

is a direct sum of regular F[S}]-modules (where F[&] and F[ji] denote thegroup rings of & and , respectively, over F).

Proof. We first prove (2). If there exists an H p such that %H and $2 areequivalent, then clearly (%iHY, %i^, and g2^ are all equivalent representationsof &. Conversely, assume that $1^ and %2 are equivalent representations of&. But, by Lemma 1.5,

Si'k = 0 5? and 8fs*|w = 0 gf,

and (2) follows. Since the number of ^-regular classes of S? is

Ao($) + (Ao(SR) - 1 ) / | $ | ,

(3) follows. Then (1) is immediate and (4) follows as in the proof of[6,KapitelV,Satz 16.13].

Finally, we state a generalization of [3, Theorem 3.4.3] which follows fromour lemmas and Clifford's theorem.

COROLLARY 1.7. Let @ = yiQ be a Frobenius group with Frobenius kernel 9and complement >. Let F be a splitting field for SSI and let % be an irreduciblerepresentation of & with Ker(g) 2 9. Then g|^ has exactly | | distinctWedderburn components.

2. Lemmas in homological algebra of groups. In this section, we derivea result (Lemma 2.6) which is needed in the proof of one of our main theorems(Theorem 3.2) as a consequence of various group homological results which areof independent interest.

LEMMA 2.1. Let K denote an arbitrary field, let & denote an arbitrary finitegroup, and let (S denote the category of right K[&]-modules (where K[&] denotesthe group ring of & over K). Then, for any objects V, W in (S and any integern ^ 1, we have:

ExtKW(V, W)9H"(&, HomK(F, W))

as abelian groups, where H o m ^ F , W) is viewed as a right &'-module with actiondefined by: if G ^ , v G V,f G Hom*(F, W), then v(fG) = ((vG~l)f)G.

Proof. By [2, XVI, 7(6)], ExtK[^(K, UomK(V, W)) * ExtK[^(V, W)for all integers n ^ 0, where K is viewed as a trivial ^- mo du le . Let 5) denotethe category of right Z[S^]-modules and let T: E > 35 denote the obvious

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"forgetful" functor. Clearly {ExtZ[^]w(Z, *) o T\ n ^ 0} is a connected sequence

of functors in S and for each object X in ,

Extz[ , ](Z, r p O ) ^ X ^ Hornet,] (K, X) ^ E x t ^ C K , X ).

Moreover, by [8, Corollary 2.2], every i[^]-injective module P is weaklyK[@]-projective and hence T(P) is weakly Z[^]-projective. Thus

Extz[,](z, r(p = o

for all n > 0 and all i[S?]-injective modules and

{Extz[ ,]B(Z, *) oT\n ^ 0}

satisfies the well-known axiomatic description of {ExtK[9]n (K, *)| n ^ 0}

(cf. [7, Chapter II I, Theorem 10.2]). Finally, for all integers l ,

Ext^f (7, 17) ^ ExtKW(K, Horn,, (7 , 17))

^ Ext*w*(Z f r (Hom*(7 , W)))

^Hn(&,HomK(Vy 17)).

LEMMA 2.2. Let V be a vector space over a field K and let & be a group ofK-linear transformations of V (acting on the right). Let F be a subfield of K suchthat \K: F\ is finite. Then the left K[&]-modules Hom F (7 , F) and Homi5 :(7 , K)are K[&]-isomorphic.

Proof. By means of standard isomorphisms of left K[&]-modules we have:

Hom F (7 , F) ^ Hom^X" K 7, F) ^ Hom*(7, HomF(K, F)).

Let /3: HomF (7 , F) > Hom x (7, HomF(i, F)) denote this isomorphism. NowHomF(i, F) is a vector space over K and

dimK(HomF(K, F)) = , Rm p. dimF(HomF(if, F)) = L ; ' = 1.

Let t: HomF(K, F) > K be a ^-isomorphism ; thus / induces an isomorphism

/*: Hom*(7, HomF(K, F))->HomK(V, K) of left i[^-modules. Finallya = ]8 o t* is the required isomorphism.

COROLLARY 2.3. L^ X denote an arbitrary field, let F be a subfield of K suchthat \K: F\ is finite and let & denote an arbitrary finite group. If Vis an arbitraryright K[ &]-module and if both F and K are viewed as trivial right & -modules, thenExtX[^]

w(7, K) ~ ExtF[^]w(7 , F) as abelian groups for any integer n ^ 1.

Proof. By Lemma 2.1, ExtKWn(V, X) ^ Hn(@, Hom*(7, K)) and

ExtF[^n(V, F)^LHn{&, Hom F (7 , F)) for all integers n ^ l where the actions

of S on the right on Ho m^ (7, K) and Hom F (7 , F) are as defined in Lemma 2.1.

However, by Lemma 2.2, HomjK:(7, K) and Hom F (7 , F) are isomorphic underthis action; henceH(&, Horn* (7 , K))^Hn(@, Hom F (7 , F)) for all integersn ^ 1, and the result follows.

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Using the same method of proof as in [4, Lemma 4], we can prove the following result.

LEMMA 2.4. Let V be a vector space of dimension n ^ 3 over afield K. Let U bea subspace of V of dimension 1 over K. Let & = S L(F / U, K) and suppose thata: & > GL(F, K) is a monomorphism such that if G &, then Ga is trivial onU and Ga induces G on V/U. Then, excluding the cases (a) n = 3, char K = 2and (b) n = 4 and \K\ 2, there exists a subspace W of V which is &-invariantsuch that V = U W.

COROLLARY 2.5. If X is a vectorspace of dimension n ^ 2 overafield K, thenyexcluding the cases (a) n = 2, char K = 2 and (b) n = 3, \K\ = 2, we haveExtK[8Ux,K))(X, K) = 0, where the action of SL(X, K) on X is the naturalaction and on K is the trivial action.

We conclude this section with a result needed in the proof of Theorem 3.2.

LEMMA 2.6. Let K and X be as in Corollary 2.5 and assume also that K is afinite field. Let F be a sub field of K and let & be a group such that

GL(X, K) 3 ^ 3 SL(X, K). Then, excluding the cases (a) and (b) above, wehave Ext^t^]1^ F) = 0, where the action of & on X is the natural action andon F is the trivial action.

Proof. Since | &: SL(X, K)\ is relatively prime to char K and Hl(SL(X, K),

H om^X , K)) = (0) by Lemma 2.1 and Corollary 2.5, we haveH^^,nomK{X,K)) = (0).

Then Extxiy]1 (X, K) = (0) follows from Lemma 2.1 and E x t ^ ^ p f , F) = (0)follows from Corollary 2.3.

3. Lower bounds for the degrees of projective representations of two

types of groups. This section contains our basic results which give lowerbounds on the degrees of non-trivial projective representations for groups containing subgroups of two specific types. These results are applied in 4 to the

groups PSL(n, q).Let ^ be a finite group and let F be a finite-dimensional vector space over a

field K. Let p: S? > GL(F, K) be a projective representation of & and letp*: & PGL(F, K) denote the p-induced group homomorphism.

Definition. The projective representation p: 2^ >GL(F, K) is said to befaithful if p*: ^ PGL(F, K) is a monomorphism.

THEOREM 3.1. Let & be a finite group which contains a subgroup 3f which is aFrobenius group with elementary abelian kernel of order qn with n ^ 1 and q pJ\

p a prime, f^ 1, and with cyclic complement of order (l/d)(qn 1) where

d = gcd{n + I, q 1}. If p: ^ * GL(V, K) is a projective representation of &such that p restricted to ^f is faithful, if p 9^ char K, and if (n, q) ^ (1, 4),(1,9), (2, 2), and (2, ),thendegp (= dim^F) Wd)(qn - 1).

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Proof. Clearly we may assume that ^ = Jf, that K is algebraically closed,and that p*: ^ ->PGL(T,i) isamonomorphism. Note that (l/d)(qn - 1) = 1if and only if(n,q) = (1,2) or(n,q) = (1 , 3) ; thus we may exclude (n,q) = (1,2)and (n, q) (1, 3). Since & has a finite representation group over K, thereexists a finite subgroup g ofGL(F, K) ofminimal order such that if

3 = Z(GL(V,K)),

then S / ( S < 3 ) = - Let $tbe the pre-image of the Frobenius kernel in g.Since fl/(gn 3 ) is abelian and g H 3 C Z ( $ ) , isnilpotent. Ifp f |g Pi 3 | ,t h e n J ? = ( g P 3 ) X < i)3, where ^ is elementary abelian oforder qn and is the-Sylow subgroup of $. Since g is a subgroup ofGL(F, K) and char i 5 p,

Lemma 1.3 applies, and we conclude that deg p ^ (l/d) (qn

1). Thus for theremainder of the proof, we assume that p\ |g P \. Here $ = 0P>($) X $ ,where $ denotes the ^-Sylow subgroup of $, 0 ^ ($) = 0P> ( g H 3) and where^/0P(% H 3 ) is an elementary abelian ^-group oforder q

n. Since % P 3 iscyclic, let 9 denote the unique subgroup of index p in 0P(% P 3 ) ; clearly31 < g. If $/5ft is non-abelian, then ($/$)' = 0P($ P 3 ) / ^ = *($/*)since $ / 0 p ( g H 3) is elementary abelian and \0P(% P 3) /9 | = . Note that2^ operates irreducibly by conjugation on its Frobenius kernel. For, theFrobenius kernel of ^ is a vector space over GF(p) oforder qn and a Frobeniuscomplement has order (l/d)(qn 1) which is relatively prime to p. By

complete reducibility and the fixed-point free action of the complement on thekernel, if & acts reducibly on its Frobenius kernel, then (l/d) (qn 1) ^ q& 1.This implies that

(qn - 1) ^ (n - l) (g - 1) = qkn+2) - qh - q+ 1 < qkn+2) - 1,

andhencen < \(n + 2) orn < 2. But then n = 1, d S 2, q2 - 1 ^ 2(q - 1),and hence q + 1 ^ 2 which is impossible. Consequently, $ acts irreducibly on^/0P(% P 3 ) , and hence Z($/STC) = 0 p ( g P 3 ) /^ . T h u s $/3t is an extra-special >-group. Let be the inverse image in g of a Frobenius complement of@. Clearly = 1 X 0

P(% C\3 ) , where 1 is a ^'-group containing

0p>(%r^) = 0p.($t). Let * = ! X 5ft; then

Cs*0P/O,(gn3)) = 0p'() X SR,

and hence */iteZ V, Sate 17.13]applies, and we conclude that (l/d)(qn 1) ^ (q*n + 1). However, gw = pfn

and since $/$# is extra-special, 2\fn. Thus

^^ - 1 ( ^ + l)(pf - 1) = *' + / _ */* - 1.

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If n ^ 2, then pfn - 1 ^ */ _ i and hence n S \(n + 2) and thenn = 2. But if ^ = 2, d| 3 and (g2 - 1) ^ d(q + 1); thus g - l g i which

implies th at g = 2, g = 3 or g = 4. Since (n, g) ^ (2, 2) and (n, q) ^ (2, 4) ,we must have q = 3. But then d = 1 and 32 1 ^ 3 + 1 which is a contradiction. Hence assume that n = 1. If g is even, d = 1 and g 1 ^ Vg + 1.Thus g2 4g + 4 ^ g, g2 4g < g, g 4 < 1, and g = 2 or g = 4. Since thecases (n, g) = (1, 2) and (n, g) = (1, 4) have been excluded, we may assumethat g is odd and then d = 2 and g 1 ^ 2(V PGL(F, K) is a monomorphism.Moreover, there exists a finite subgroup g of GL(F, X) of minimal

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PROJECTIVE REPRESENTATIONS OF PSL(W, q) 99

order such that g / ( g H 3) = W2. Let $ be the pre-image of W in g;as above, $ is nilpotent. If | | g H 31, then $ = ( g H ^ X ? , where $ is

the ^>-Sylow subgroup of $ and is isomorphic as a group to W and where0*'($) = g

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4. Lower bounds for the degrees of projective representations of the

groups PSL(w, q). We now proceed to show how the results of 3 can be

applied to obtain lower bounds for the degrees of projective representations ofthe finite groups PSL(w, q), and we give a new proof of a classical result(Theorem 4.5).

LEMMA 4.1. Let n ^ 2 be an integer, let q = pf , where p is a prime integer andfis a positive integer, let F denote afield of q elements, and let d = gcd{n, q 1}.Then:

(a) PSL(n, q) contains a subgroup isomorphic to the "natural" semi-direct product W2, where W denotes a vectorspace of dimension n 1 over F andwhere 8 = {X e GL(W, F)\ detX 6 (P

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PROJECTIVE REPRESENTATIONS OF PSL(W, q) 101

THEOREM 4.2. Using the notation above (n ^ 2), if (n, q) ^ (2, 4) , (2, 9),

(3 , 2), (3, 4), then the degree of any non-trivial projective representation p ofPSL(n, q) overany fieldK with char K ^ p is atleast (l/d)(qn~1 1).

Proof. Since (\/d)(qn-1 - 1) = 1 ifand only if(n,q) = (2, 2) or (n,q) = (2, 3),we may also assume tha t (n,q) ^ (2, 2) and (n,q) 9e (2, 3). But then PSL(n, q) issimple and so the p induced group homomorphism p*: PSL(w, g) > PGL(^, i),where n = deg p, is a monomorphism since p is non-trivial. The theorem nowfollows from Theorem 3.1 and Lemma 4.1 (b).

A similar proof using Theorem 3.2 and Lemma 4.1 (a) yields the followingresult.

THEOREM 4.3. If n ^ 3 and if the cases n = 3, p = 2 and n = 4, q = 2 areexcluded, then the degree of any non-trivial projective representation p ofPSL(^, q)overany fieldK with char K 9e p is atleast qn~1 1.

For projective representations of PSL(w, q) in the characteristic p case wehave the following result.

THEOREM 4.4. Let n ^ 2 and q = pf be as above. If (n, q) ^ (2, 2) and(n, q) 9e- (2, 3) , then the degree of any non-trivial projective representation ofPSL(w, q) overany fieldofcharacteristic p is atleast n.

Proof. Let p denote a non-trivial projective representation of PSL(w, q) ofdegree n over a field K of characteristic p. Clearly we may assume that K isalgebraically closed. Since PSL(?z, q) has a finite representation group over K,we may assume that K is the algebraic closure of GF(^>). Since PSL(w, q) isfinite, we may assume that p takes values in GL{n, K), where K is a finite fieldof characteristic p. Since PSL(n, q) is simple, p induces a group monomorphismp*: PSL(w, q) > PGL(?z, j). But the ^-Sylow subgroups of PSL(n, q) andPGL(r, .K) have classes n 1 and w 1, respectively (cf. [6, Kapitel III,Sate 16.3]). Hence n ^ n.

To conclude, we give a new proof of the following well-known result(cf. [6, Kapitel II, Satz 6.14]).

THEOREM 4.5. Let Wi 2 and n2 2 e integers and let q\ = ^ i/ x awd

52 = ^2/2> where p\ and pi are prime integers andf\ and f2 are positive integers.If PSL(^i, qi) =L PSL(n2, #2) with (wi, gi) 9^ (w2, #2), ^0 we ^ e 0^e of thefollwing two cases:

PSL(2, 4) ^ PSL(2, 5) or PSL(2, 7) ^ PSL(3, 2).

Proof. Since PSL(2, 2) and PSL(2, 3) are the only solvable groups under

consideration and have different orders, we may assume that (nit qt) 9

e

- (2, 2)and {nu qt) 9* (2, 3) for i 1, 2. If pi p2l then Theorem 4.4 implies thatn\ ^ n2 and n2 ^ Wi. Hence, in this case, n\ n2 and, comparing the orders of-Sylow subgroups, we conclude that q\ = q2. Thus we may assume that

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pi 7e P2. First suppose that n = n2 = n. Ifn 3, then pi ^ 2 or p2 ^ 2 and

Theorem 4.3 implies that n ^ qn

~

l

- U 3

7

*-

1

- 1. Since 37

*-

1

- 1 > n for ailn ^ 3, we may assume th at n = 2. Suppose th at pi 2; thengcd{2, gi 1} = 1and 2 ^ gi 1 by Theorem 4.2. Hence qi = 4; but since

|PSL( 2,4)| ^ | PSL(2,9 )|,

we must have q2 ^ 9. Applying Theorem 4.2 again, we obtain 2 ^ %(q2 1) orq2 S 5. Hence q2 = 5; but, as is well known, PSL(2, 4) ^ PSL(2, 5) (cf. [6,Kapitel II, 5a/s 6.14]). Suppose now that ^ 1 ^ 2 ^ ^2. Then either qt = 9 or4 g< - l f o r i = 1,2 by Theorem 4.2. But |PSL(2, 5) | ^ |PSL(2, 9) | and sothis case is excluded. Thus we may now assume that 2 ^ ni < n2. If p2 ^ 2,

then ni ^ g2W2_1 1 ^ 3W2_1 1 > w2 > ^i (using Theorem 4.3) and hencep2 = 2 and pi is odd. If n2 = 4, then gcd{n2, q2 1} = 1 and Theorems 4.2and 4.3 imply: if n2 ^ 4, then wi ^ g2

W2_1 - l e 2n2_1 1 > w2 > Wi, whichis impossible. Hence w2 = 3 and ni = 2. If g2 ^ 8, then

2 ^ ^ Tx (g22 - l) (Z2 + l

gcd{3, q2 - 1}

by Theorem 4.2; thus q2 = 2 or q2 = 4. Since|PSL(3, 2)| ^ |PSL(2, 9)| * |PSL(3, 4)|,

we have qi ^ 9. Then 3 ^ J(gi 1) or ci ^ 7. Hence #i = 5 or 7 and, bycomparing orders, we must have q2 = 2 and qi = 7. But, here again,PSL(2, 7) ^ PSL(3 , 2) [6, Kapitel II, Sate 6.14].

REFERENCES

1. R. Brauer, Zur Darstellungstheorie der Gruppen endlicher Ordnung, Math. Z. 63 (1956),406-444.

2. H. Cartan and S. Eilenberg, Homologuai algebra (Princeton Univ. Press, Princeton, N.J.,1956).

3. D. Gorenstein, Finite groups (Harper and Row, New York, 1968).

4. D. G. Higman, Flag-transitive collineation groups of finite projective spaces, Illinois J. Math. 6(1962), 434-446.

5. C. Hering, On transitive linear groups (in preparation).6. B. Huppert, Endliche Gruppen. I (Springer-Verlag, Berlin, 1967).7. S. MacLane, Homology (Springer-Verlag, Berlin, 1963).8. D. S. Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700-712.

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cjm1971v23.0090-0102