On finite minimal non-nilpotent groups

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 12, Pages 3455–3462 S 0002-9939(05)07996-7 Article electronically published on June 8, 2005 ON FINITE MINIMAL NON-NILPOTENT GROUPS A. BALLESTER-BOLINCHES, R. ESTEBAN-ROMERO, AND DEREK J. S. ROBINSON (Communicated by Jonathan I. Hall) Abstract. Acritical group for a class of groups Xis a minimal non-X-group. The critical groups are determined for various classes of finite groups. As a consequence, a classification of the minimal non-nilpotent groups (also called Schmidt groups) is given, together with a complete proof of Gol’fand’s theorem on maximal Schmidt groups. 1. Introduction Given a class of groups X,wesaythatagroupGis a minimal non-X-group, or an X-critical group,ifG/∈X, but all proper subgroups of Gbelong to X.Itis clear that detailed knowledge of the structure of minimal non-X-groups can provide insight into what makes a group belong to X. All groups considered in this paper are finite Minimal non-X-groups have been studied for various classes of groups X.For instance, minimal non-abelian groups were analysed by Miller and Moreno [10], while Schmidt [14] studied minimal non-nilpotent groups. The latter are now known as Schmidt groups.Itˆo [9] considered the minimal non-p-nilpotent groups for pa prime, which turn out to be just the Schmidt groups. Finally, the third author [12] characterised the minimal non-T-groups (T-groups are groups in which normality is a transitive relation). He also characterised in [13] the minimal non-PST -groups, where a PST-group is a group in which Sylow permutability is a transitive relation. The aim of this paper is to give more precise information about the structure of Schmidt groups and show how to construct them in an efficient way. As a consequence of our study, a new proof of a classical theorem of Gol’fand is given. Our approach depends on the classification of critical groups for the class of PST- groups given in [13]. Recall that a subgroup His said to be Sylow-permutable,or S-permutable, in a group Gif Hpermutes with every Sylow subgroup of G.We mention a similar class Yp, which was introduced in [2]. If pis a prime, a group Gbelongs to the class Ypif Genjoys the following property: if Hand Kare psubgroups of Gsuch that His contained in K,thenHis S-permutable in NG(K). Clearly every PST-group is a Yp-group. There is a close relation between the class of groups just introduced and pnilpotence, as in shown by the following result, which was proved in [2, Theorem 5]. Received by the editors December 12, 2003 and, in revised form, July 16, 2004. 2000 Mathematics Subject Classification. Primary 20D10. This work was supported by Proyecto BFM2001-1667-C03-03 (MCyT) and FEDER (European Union). c 2005 American Mathematical Society 3455 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3456 A. BALLESTER-BOLINCHES ET AL. Theorem 1. AgroupGis a Yp-group if and only if either it is p-nilpotent or it has an abelian Sylow p-subgroup Pand every subgroup of Pis normal in NG(P). Ourfirstmainresultis: Theorem 2. The minimal non-Yp-groups are just the minimal non-PST-groups with a non-trivial normal Sylow p-subgroup. Such groups are of the types described in Ito IV below. Let p and q be distinct primes. Type I: G=[P]Q,whereP=a, bis an elementary abelian group of order p2,Q=zis cyclic of order qr,withqa prime such that qfdivides p−1, qf>1and r≥f,andaz=ai,bz=bij,whereiis the least positive primitive qf-th root of unity modulo pand j=1+kqf−1,with0<k<q. Type II: G=[P]Q,whereQ=zis cyclic of order qr>1,withqaprime not dividing p−1and Pan irreducible Q-module over the field of pelements with centralizer zqin Q. Type III: G=[P]Q,whereP=a0,a 1,...,a q−1is an elementary abelian p-group of order pq,Q=zis cyclic of order qr,withqa prime such that qfis the highest power of qdividing p−1and r>f. Define az j=aj+1 for 0≤j<q−1and az q−1=ai 0,whereiis a primitive qf-th root of unity modulo p. Type IV: G=[P]Q,wherePis a non-abelian special p-group of rank 2m, the order of pmodulo qbeing 2m,Q=zis cyclic of order qr>1,z induces an automorphism in Psuch that P/Φ(P)is a faithful irreducible Q-module, and zcentralizes Φ(P).Furthermore,|P/Φ(P)|=p2mand |P|≤pm. Since a group is a soluble PST-group if and only if it belongs to Ypfor all primes p[2, Theorem 4], Theorem 2 may be regarded as a local approach to the third author’s classification of minimal non-PST -groups [13]. An interesting consequence of Theorem 2 is the following classification of Schmidt groups. In order to describe the classification, we must introduce one further type of group: Type V: G=[P]Q,whereP=ais a normal subgroup of order p,Q=z is cyclic of order qr>1, and az=ai,whereiis the least primitive q-th root of unity modulo p. Our main result can now be stated as: Theorem 3. The Schmidt groups are exactly the groups of Type II,TypeIV and Type V. Our next result shows that p-soluble groups with Sylow p-subgroups isomorphic to a normal subgroup of a minimal non-Yp-group have a restricted structure. Theorem 4. Let Gbe a p-soluble group with a Sylow p-subgroup P.IfPis isomorphic to a non-trivial normal Sylow subgroup of a minimal non-Yp-group, then Ghas p-length 1. In [4] Gol’fand stated the following result: Theorem 5. Let pand qbe distinct primes, let rbe a given positive integer, and let abe the order of pmodulo q. Then there is a unique minimal non-p-nilpotent group G0of order pa0qr,wherea0=aif ais odd and a0=3a/2if ais even, such License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON FINITE MINIMAL NON-NILPOTENT GROUPS 3457 that all minimal non-p-nilpotent groups of order ptqrare isomorphic to quotients of G0by central subgroups. Only a sketch of a proof of this theorem is given in Golfand’s article. In Section 3, we show how to construct the Schmidt groups of Gol’fand, and we also give a complete proof of Theorem 5. We remark that R´edei [11] has given another construction of the Schmidt groups of maximum order. 2. Proofs of Theorems 2, 3 and 4 Proof of Theorem 2.Assume that Gis a minimal non-Yp-group and let Pbe a Sylow p-subgroup of G.SinceGdoes not belong to Yp, there exist subgroups H and Kof Psuch that H≤Kand His not S-permutable in NG(K). Consequently there is an element z∈NG(K) such that zdoes not normalise H.Hereitcanbe assumed that zhas order qrfor some prime q=p.ThenG=Kzbecause Gis a minimal non-Yp-group. This implies that K=Pis a normal Sylow p-subgroup of Gand Q=zis a cyclic Sylow q-subgroup of G.ThenGis not a PST -group, yet every proper subgroup has Ypand Yq, and thus is a PST-group by [2]. Conversely, if Gis a minimal non-PST -group, then Gdoes not have Ypfor some prime p. Since all its proper subgroups satisfy Yp, the group Gis a minimal non- Yp-group. The classification of minimal non-PST-groups given in [13] completes the proof. (Note that the groups of Types IV and V of [13] are both of Type IV above.)  Proof of Theorem 3.Let Gbe a minimal non-nilpotent group. Then Gis a minimal non-p-nilpotent group for some prime p. Suppose that Gis not a Yp-group, so that Gis a minimal non-Yp-group. By Theorem 2, the group Gis of one of Types I–IV. By examining the group structure, we see that groups of Type I and III are not minimal non-p-nilpotent. Therefore Gmust be of Type II or IV. Assume now that Gbelongs to Yp. Then by [1, Theorem A] and [3, VII, 6.18], the p-nilpotent residual Pof Gis an abelian minimal normal Sylow subgroup which is complemented in Gby a cyclic Sylow q-subgroup Q.MoreoverQnormalises each subgroup of P. This implies that Pis cyclic of order p,sayP=a. In addition, az=aifor some 0 <i<pand zqcentralizes a. This implies that imust be a primitive q-th root of unity modulo pand, by taking a suitable power of zas a generator of Q, we can assume that iis the least such positive integer. Hence Gis of Type V.  Proof of Theorem 4.Assume that Gis a p-soluble group with p-length >1andG has least order subject to possessing a Sylow p-subgroup Pwhich is isomorphic to a non-trivial normal Sylow subgroup of a Schmidt group. By [6, VI, 6.10], we conclude that Pis not abelian. Thus Pis a Sylow p-subgroup of a group of Type IV in Theorem 2. By minimality of order Op(G)=1andO p (G)=G.In addition, since the class of groups of p-length at most 1 is a saturated formation, we have Φ(G) = 1 and hence Ghas a unique minimal normal subgroup which is an elementary abelian p-group. Let D=O p(G); then Dis a non-trivial elementary abelian group and CG(D)=D.MoreoverΦ(P)=Z(P)≤Dand so P/D is elementary abelian. Let Tbe the subgroup defined by T/D =O p(G/D). Since P/D is an elementary abelian p-group, G/D has p-length at most 1 by [6, VI, 6.10]. It follows that (T/D)(P/D) is a normal subgroup of G/D. Therefore TP is a normal subgroup of License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3458 A. BALLESTER-BOLINCHES ET AL. G. Assume that TP is a proper subgroup of G.NowO p(TP)≤Op(G)=1,so Pis a normal subgroup of TP and hence of G, a contradiction which shows that G=TP. Assume now that P/D is a non-cyclic elementary abelian group. By [8, X, 1.9], we have T/D =CT/D(xD)|xD ∈P/D,xD =D.Letx∈P\D.SinceP/D centralizes xD,wehaveP/D ≤NG/DCT/D(xD).LetTx/D =C T/D(xD). Assume that PTx=G;thenTx=Tis a normal subgroup of Gand thus Op(G/D)=Tx/D. This implies that xD/D ≤Z(G/D)andxDis a normal p-subgroup of G,sothat xDis contained in D, a contradiction. Consequently PTxis a proper subgroup of Gfor all 1 =xD ∈P/D. Hence PTxhas p-length at most 1 by minimality of G. Since CG(D)=Dand Op(PTx) centralizes D, we conclude that Op(PTx)=1. Therefore Pis a normal subgroup of PTx, which shows that Tnormalizes Pand thus Pis a normal subgroup of G. This contradiction shows that P/D is cyclic. Since Phas class 2, we see from [7, IX, 5.5] that, if p>3, then Ghas p-length at most 1. Therefore p≤3. Let Xbe a minimal non-Yp-group such that Pis a Sylow p-subgroup of X.NotethatP/Φ(P) is an irreducible X-module. In particular D, the subgroup of the previous paragraphs, is not normal in Xand so P=DDg for some g∈X.SinceDis abelian, D∩Dg≤Z(P)=Φ(P), and it follows that P/Φ(P) has order p2. This implies that Pis an extra-special group of order p3.If p= 2, then, since CG(D)=D,weseethatGmust be a symmetric group of degree 4. Hence Pis dihedral of order 8, which cannot lead to a group of Type IV since Aut(P) is a 2-group. Hence p= 3. But a non-abelian group of order 33cannot occur as the normal Sylow 3-subgroup of a Schmidt group, because the only prime divisor of 32−1 is 2 and the order of 3 modulo 2 is 1. This contradiction completes the proof of the theorem.  3. The construction of Gol’fand’s groups and a proof of Gol’fand’s theorem We begin by constructing groups of Type IV with a Sylow p-subgroup Pof order p3mand |P/Φ(P)|=p2m. These groups were constructed in [13] by a different method, but the present approach is more convenient when p=2.Wewillusethe following result on linear operators. Lemma 6. Let pbeaprimeandletrbe a positive integer such that gcd(p, r)= 1.Letβbe a linear operator of order puron a vector space Vover the field of p-elements, where uis a non-negative integer. If βhas irreducible minimum polynomial f,thenβpualso has minimum polynomial f. Proof. Let gbe the minimum polynomial of βpu.Nowf(βpu)=f(β)pu=0,so that gdivides f.Sincefis irreducible, f=g. Construction 7. Let pand qbe distinct primes such that the order of pmodulo qis 2m,m≥1. Let Fbethefreegroupwithbasis{f0,f 1,...,f 2m−1}.Write R=FFpand R∗=[F,R]Rp.ThenF/R is an elementary abelian p-group of order p2mand H=F/R∗is a p-group such that R/R∗=Φ(H)isanelementary abelian p-group contained in Z(H). Moreover His a non-abelian group because an extra-special group of order p2m+1 is an epimorphic image of H. Denote by githe image of fiunder the natural epimorphism of Fonto H= F/R∗,0≤i≤2m−1. Since Hhas class 2, we know that Φ(H) is generated by all [gi,g j], with i<j,andgp i. Therefore Φ(H) has dimension as GF(p)-vector space License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON FINITE MINIMAL NON-NILPOTENT GROUPS 3459 at most 1 2(2m(2m−1))+2m=m(2m+1). Assume that the dimension is less than m(2m+ 1). Then there exists an element r= j (fp j)λj j<k [fj,f k]µjk ∈R∗ with some λjor µjk not divisible by p. It is clear that p|λjfor all jsince FpF/F  is a free abelian group with basis {fp jF|0≤j≤2m−1}. Suppose that pµik for some i<kand let ρibe the endomorphism of Fdefined by fρi i=f2 i,fρi l=fl for l=i.ThenrρiR∗=R∗and so rρir−1R∗=R∗. This implies that w= j<i [fj,f i]µji  i<l [fi,f l]µil ∈R∗. On the other hand, by applying ρkwe find that wρkw−1R∗=[fi,f k]µik R∗=R∗. Since pµik, it follows that µik has an inverse modulo p. This means that [fi,f k]∈ R∗. Now since permutations of the generators of Finduce endomorphisms in F and R∗is fully invariant, it follows that F≤R∗and His abelian, a contradiction. Therefore Φ(H) has dimension m(2m+1)andso|Φ(H)|=pm(2m+1). Let f(t)=c0+c1t+···+c2m−1t2m−1+t2mbe an irreducible factor of the cyclotomic polynomial of order qover GF(p)andletαbe the endomorphism of Fgiven by fα i=fi+1 for 0 ≤i≤2m−2, fα 2m−1=f−c0 0f−c1 1···f−c2m−1 2m−1.Since R∗is a fully invariant subgroup of F, it follows that αinduces an endomorphism βon H=F/R∗. In turn, βinduces an automorphism ¯ βon H/Φ(H). Since H/Φ(H)=H/Φ(H)¯ β≤HβΦ(H)/Φ(H), it follows that H=HβΦ(H), whence H=Hβ.Consequentlyβis an automorphism of H. It is clear that βinduces the linear operator ¯ β, with minimum polynomial f, on the vector space H/Φ(H). Now by [6, III, 3.18], we conclude that βqhas order pufor a some uand hence βhas order puq. By Lemma 6, there is a GF(p)-basis {g 0,g 1,...,g 2m−1}of H/Φ(H), where g i=giΦ(H), such that g¯ βpu i=g i+1 for 0≤i≤2m−2andg¯ βpu 2m−1=g−c0 0g−c1 1···g−c2m−1 2m−1. Hence we can replace βby βpuand assume without loss of generality that βhas order q. It follows that Φ(H)isaGF(p)T-module, where T=βis a cyclic group of order q.ByMaschke’sTheoremΦ(H) is a direct sum of irreducible T-modules. Let Nbe the sum of all non-trivial irreducible submodules in the direct decomposition and write P=H/N. It is clear that Nis β-invariant and therefore βinduces an automorphism γof order qin P.LetQ=zbe a cyclic group of order qracting on Pvia z→ γ.LetG=[P]Qbe the corresponding semidirect product. It is easily checked that Gis a Schmidt group. Next we show that Phas order p3m. From Theorem 3 we see that Φ(P) has order at most pm,where |P/Φ(P)|=p2m. On the other hand, |Φ(H)|=pm(2m+1),andNhas order a power of p2mbecause every faithful irreducible β-module over GF(p) has dimension 2m. Therefore |Φ(P)|=pm. Remark 8.In the group of Construction 7, we may assume that ¯gz 2m−1=¯g−c0 0¯g−c1 1 ···¯g−c2m−1 2m−1,where¯gi=giN. Proof. We know that ¯gz 2m−1=¯g−c0 0¯g−c1 1···¯g−c2m−1 2m−1¯w,where ¯w∈Φ(P). Since f(t) is irreducible, 1 is not a root of f(t) and it follows that c=c0+c1+···+c2m−1+1 ≡ 0 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3460 A. BALLESTER-BOLINCHES ET AL. (mod p). Consequently there exists an integer dsuch that cd ≡−1(modp). Put ¯w0=¯wdand consider the automorphism δof Pdefined by ¯gδ i=¯gi¯w0for 0 ≤i≤ 2m−1. If we write γ0=δγδ−1, it is easily checked by an elementary calculation that ¯gγ0 i=¯gi+1 for 0 ≤i≤2m−2, and ¯gγ0 2m−1=¯g−c0 0¯g−c1 1···¯g−c2m−1 2m−1.Letz0be a cyclic group of order qr,withz0acting on Pvia z0→ γ0.Sincez0and zare conjugate in Aut(P), it follows by [3, B, 12.1] that the groups Pzand Pz0are isomorphic.  Remark 9.The group in Construction 7 does not depend on the choice of irreducible factor f(t). Proof. Assume that the group G1=[P1]z1has been constructed by using another irreducible factor g(t) of the cyclotomic polynomial of order qover GF(p). Since Gand G1have the same order, it will be enough to find a set of generators of G1for which the relations of Ghold. Since zcentralizes Φ(P)andz1centralizes Φ(P1), we have G/Φ(P)∼ =[P/Φ(P)]zand G1/Φ(P1)∼ =[P1/Φ(P1)]z1.But P/Φ(P)andP1/Φ(P1) are faithful irreducible modules for a cyclic group of order q. Therefore [P/Φ(P)](z/zq) is isomorphic to [P1/Φ(P1)](z1/zq 1)by[3,B, 12.4]. Let φbe an isomorphism between these groups. Then it is clear that φ induces an isomorphism ψbetween G/Φ(P)andG1/Φ(P1). Let ¯ hi=hiΦ(P), 0 ≤i≤2m−1. Put ¯ ki=¯ hψ iand ¯u=¯zψ.Weshowhowto extend the isomorphism ψto an isomorphism between Gand G1.Inordertodoso, we choose representatives kiof ¯ kiand uof ¯usuch that the order of uis qr.There is no loss of generality in assuming that ku i=ki+1 for 0 ≤i≤2m−2. Indeed, if ku i=ki+1wi+1 with wi+1 ∈Φ(P1), then k i=kiwi···w1for 1 ≤i≤2m−1, k 0=k0 are representatives of ¯ kiand k i u=k i+1 for 1 ≤i≤2m−1 because ucentralizes Φ(P1). By using the same argument as in Remark 8, we may also assume that ku 2m−1=k−c0 0k−c1 1···k−c2m−1 2m−1. Therefore Gand G1satisfy the same relations and by Von Dyck’s theorem they are isomorphic.  Remark 10.In Construction 7, it is not necessary to assume that βhas order q.Indeed, it can be proved that βqfixes all elements of Φ(H) and that the automorphism γinduced by βin H/N has order q. Gol’fand’s result (Theorem 5) can be recovered with the help of Construction 7 and Theorem 3. Proof of Theorem 5.Let pand qbe distinct primes and let abe the order of p modulo q.Thenais the dimension of each non-trivial irreducible module for a cyclic group of order qover GF(p). Assume that ais odd. Then every Schmidt group Gwith a normal Sylow p-subgroup Psuch that |P/Φ(P)|=pais of Type II or Type V. Then the theorem holds in this case because all Schmidt groups of the same type with isomorphic Sylow q-subgroups are actually isomorphic. Assume now that ais even, with say a=2m. Then we are dealing with Schmidt groups of Type II or Type IV. Let G0be the group of Construction 7. Then |G0|=p3mqrand |P0/Φ(P0)|=p2m,whereP0is a normal Sylow p-subgroup of G0. It is clear that G0/Φ(P0) is a Schmidt group of Type II. Therefore, if Gis a Schmidt group of Type II with order ptqrand a normal Sylow p-subgroup, then G∼ =G0/Φ(P0)andΦ(P0)≤Z(G0). Consequently, we need only show that all Schmidt groups of Type IV and order ptqr,t≤3m, which have a normal Sylow p-subgroup are isomorphic to quotients of G0by central subgroups. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON FINITE MINIMAL NON-NILPOTENT GROUPS 3461 Let Gbe a Schmidt group of Type IV and order ptqrwith a normal Sylow psubgroup ¯ P.ThenG0/Φ(P0)andG/Φ(P) are isomorphic. Let us choose generators zand ¯zof Sylow q-subgroups Qof G0and Qof Gsuch that the minimum polynomials of the actions of zon P0/Φ(P0)and¯zon P/Φ(P) coincide. Also choose generators g0,g1,...,g2m−1of the Sylow p-subgroup P0of G0and generators ¯g0, ¯g1,...,¯g2m−1of the Sylow p-subgroup Pof Gsuch that gz j=gj+1 and ¯g¯z j=¯gj+1 for 0 ≤j≤2m−2. Since Φ(P0)=P 0and Φ(P)=P,andbothP0and Phave class 2, the subgroup Φ(P0) can be generated by the commutators [gi,g j], while Φ(P) is generated by the commutators [¯gi,¯gj]. On the other hand, if ui=[g0,gzi 0], we have ui=uzk i=[gk,gzi k]. It is easy to see that ui=[g0,gzi 0]=[gzq 0,gzi 0]=u−1 q−i. Observe that qis odd since 2mdivides q−1: write q=2s+ 1. By definition of the giand ui, and use of the minimum polynomial of the action of zon P0/Φ(P0), it may be shown that for l≥1, us+m+l=u−c0 s−m+lu−c1 s−m+l+1 ...u −c2m−2 s+m+l−2u−c2m−1 s+m+l−1. Now this formula and the relations ui=u−1 q−iallow us to show by induction that each us+m+lcan be expressed in terms of elements of the set B={us−m+l, us−m+2,...,u s}.SinceΦ(P0) has dimension mover GF(p), this expression is unique. It follows that each ujcan be uniquely expressed in terms of the elements of B, and so this is also true for each generator of Φ(P0). Thesameargument shows that the generators of Φ(P) have a similar unique expression subject to the same relations. The arguments of Remark 9 allow us to assume that gz 2m−1=g−c0 0g−c1 1···g−c2m−1 2m−1and ¯g¯z 2m−1=¯g−c0 0¯g−c1 1···¯g−c2m−1 2m−1. Consequently, all relations of G0are satisfied by G. By Von Dyck’s theorem, it follows that Gis an epimorphic image of G0by a central subgroup of G0. References [1] A. Ballester-Bolinches and R. Esteban-Romero. Sylow permutable subnormal subgroups of finite groups II. Bull. Austral. Math. Soc., 64(3):479–486, 2001. MR1878899 (2002k:20035) [2] A. Ballester-Bolinches and R. Esteban-Romero. Sylow permutable subnormal subgroups of finite groups. J. Algebra, 251(2):727–738, 2002. MR1917388 (2003f:20031) [3]K.DoerkandT.Hawkes.Finite Soluble Groups. De Gruyter Expositions in Mathematics. 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Debrecen, 4:303–324, 1956. MR0078998 (18:12f) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 3462 A. BALLESTER-BOLINCHES ET AL. [12] D. J. S. Robinson. Groups which are minimal with respect to normality being intransitive. Pacific J. Math., 31:777–785, 1969. MR0258943 (41:3588) [13] D. J. S. Robinson. Minimality and Sylow-permutability in locally finite groups. Ukr. Math. J., 54(6):1038–1049, 2002. MR1956642 (2003k:20054) [14] O. J. Schmidt. ¨ Uber Gruppen, deren s¨amtliche Teiler spezielle Gruppen sind. Mat. Sbornik, 31:366–372, 1924. Departament d’` Algebra, Universitat de Val` encia, Dr. Moliner, 50, E-46100 Burjassot, Val` encia, Spain E-mail address:[email protected] Departament de Matem` atica Aplicada, Universitat Polit` ecnica de Val` encia, Cam´ ıde Vera, s/n, E-46022 Val` encia, Spain E-mail address:[email protected] Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801 E-mail address:[email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use