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Abstract Algebra: Supplementary Lecture Notes JOHNA.BEACHY NorthernIllinoisUniversity 1995 Revised,1999,2006 ii Toaccompany AbstractAlgebra,ThirdEdition byJohnA.BeachyandWilliamD.Blair ISBN1–57766–434–4,Copyright2006 WavelandPress,Inc. 4180ILRoute83,Suite101 LongGrove,Illinois60047 847/634-0081 Copyright©2006,1999,1995byJohnA.Beachy Permission is granted to copy this document in electronic form, or to print it for personaluse,undertheseconditions: itmustbereproducedinwhole; itmustnotbemodifiedinanyway; itmustnotbeusedaspartofanotherpublication. FormattedJanuary12,2007,atwhichtimetheoriginalwasavailableat: http://www.math.niu.edu/∼beachy/abstract_algebra/ Contents PREFACE v 7 STRUCTUREOFGROUPS(cont’d) 485 7.8 NilpotentGroups . . . . . . . . . . . . . . . . . . . . . . . . . . 485 7.9 SemidirectProducts . . . . . . . . . . . . . . . . . . . . . . . . . 488 7.10 ClassificationofGroupsofSmallOrder . . . . . . . . . . . . . . 496 BIBLIOGRAPHY 502 INDEX 503 iii iv CONTENTS PREFACE These notes are provided as a supplement to the book Abstract Algebra, Third Edition,byJohnA.BeachyandWilliamD.Blair,WavelandPress,2006. Thenotesareintendedfortheuseofgraduatestudentswhoarestudyingfrom ourtextandneedtocoveradditionaltopics. JohnA.Beachy October,2006 v vi PREFACE Chapter 7 STRUCTURE OF GROUPS (cont’d) 7.8 Nilpotent Groups We now define and study a class of solvable groups that includes all finite abelian groupsandallfinite p-groups. Thisclasshassomeratherinterestingproperties. 7.8.1 Definition. For a group G we define the ascending central series Z (G) ⊆ 1 Z (G) ⊆ ··· ofG asfollows: 2 Z (G)isthecenter Z(G)ofG; 1 Z (G)istheuniquesubgroupofG with Z (G) ⊆ Z (G)and Z (G)/Z (G) = 2 1 2 2 1 Z(G/Z (G)). 1 Wedefine Zi(G)inductively,sothat Zi(G)/Zi−1(G) = Z(G/Zi−1(G)). ThegroupGiscallednilpotentifthereexistsapositiveintegernwith Z (G) = n G. Wefirstnotethatanyabeliangroupisnilpotent. Wenextnotethatanynilpotent groupissolvable,sincethefactorgroups Zi+1(G)/Zi(G)areabelian. Wealsonote that these classes are distinct. The proof of Theorem 7.6.3 shows that any finite p-group is nilpotent, so the group of quaternion units provides an example of a groupthatisnilpotentbutnotabelian. Thesymmetricgroup S issolvable,butitis 3 notnilpotentsinceitscenteristrivial. WewillshowthattheconverseofLagrange’stheoremholdsfornilpotentgroups. Recall that the standard counterexample to the converse of Lagrange’s theorem is the alternating group A , which has 12 elements but no subgroup of order 6. We 4 485 486 CHAPTER7. STRUCTUREOFGROUPS(CONT’D) notethat A isanotherexampleofasolvablegroupthatisnotnilpotent. Itfollows 4 fromTheorem7.4.1,thefirstSylowtheorem,thatanyfinite p-grouphassubgroups ofallpossibleorders. Thisresultcanbeeasilyextendedtoanygroupthatisadirect productof p-groups. ThustheconverseofLagrange’stheoremholdsforanyfinite abelian group, and this argument will also show (see Corollary 7.8.5) that it holds foranyfinitenilpotentgroup. Wefirstprovethatanyfinitedirectproductofnilpotentgroupsisnilpotent. 7.8.2Proposition. IfG ,G ,...,G arenilpotentgroups,thensois 1 2 n G = G ×G ×···×G . 1 2 n Proof. Itisimmediatethatanelement(a ,a ,...,a )belongstothecenter Z(G) 1 2 n ofG ifandonlyifeachcomponenta belongsto Z(G ). Thusfactoringout Z(G) i i yields G/Z(G) = (G /Z(G ))×···×(G /Z(G )) . 1 1 n n Usingthedescriptionofthecenterofadirectproductofgroups,weseethat Z (G) = Z (G )×···× Z (G ) , 2 2 1 2 n andthisargumentcanbecontinuedinductively. Ifmisthemaximumofthelengths oftheascendingcentralseriesforthefactors G ,thenitisclearthattheascending i centralseriesforG willterminateatG afteratmostm terms. (cid:50) Thefollowingtheoremgivesourprimarycharacterizationofnilpotentgroups. WefirstneedalemmaaboutthenormalizerofaSylowsubgroup. 7.8.3Lemma. If P isaSylow p-subgroupofafinitegroupG,thenthenormalizer N(P)isequaltoitsownnormalizerinG. Proof. SincePisnormalinN(P),itistheuniqueSylow p-subgroupofN(P). Ifg belongstothenormalizerof N(P),then gN(P)g−1 ⊆ N(P),so gPg−1 ⊆ N(P), whichimpliesthatgPg−1 = P. Thusg ∈ N(P). (cid:50) 7.8.4Theorem. ThefollowingconditionsareequivalentforanyfinitegroupG. (1)G isnilpotent; (2)nopropersubgroup H ofG isequaltoitsnormalizer N(H); (3)everySylowsubgroupofG isnormal; (4)G isadirectproductofitsSylowsubgroups. 7.8. NILPOTENTGROUPS 487 Proof. (1)implies(2): Assumethat G isnilpotentand H isapropersubgroupof G. Withthenotation Z (G) = {e},letnbethelargestindexsuchthat Z (G) ⊆ H. 0 n Then there exists a ∈ Zn+1(G) with a (cid:54)∈ H. For any h ∈ H, the cosets aZn(G) andhZ (G)commuteinG/Z (G),soaha−1h−1 ∈ Z (G) ⊆ H,whichshowsthat n n n aha−1 ∈ H. Thusa ∈ N(H) − H,asrequired. (2) implies (3): Let P be a Sylow p-subgroup of G. By Lemma 7.8.3, the normalizer N(P) is equal to its own normalizer in G, so by assumption we must have N(P) = G. Thisimpliesthe P isnormalinG. (3)implies(4): Let P , P ,..., P betheSylowsubgroupsofG,corresponding 1 2 n toprimedivisors p , p , ..., p of|G|. Wecanshowinductivelythat P ···P ∼= 1 2 n 1 i P ×···×P fori = 2,...,n. Thisfollowsimmediatelyfromtheobservationthat 1 i (P1···Pi)∩Pi+1 = {e}becauseanyelementin Pi+1hasanorderwhichisapower of pi+1, whereas the order of an element in P1 ×···× Pi is p1k1··· piki, for some integersk ,...,k . 1 n (4) implies (3): This follows immediately from Proposition 7.8.2 and the fact thatany p-groupisnilpotent(seeTheorem7.6.3). (cid:50) 7.8.5Corollary. LetG beafinitenilpotentgroupofordern. Ifm isanydivisorof n,thenG hasasubgroupoforderm. Proof. Letm = pα1··· pαk betheprimefactorizationofm. Foreachprimepower 1 k α α p i, thecorresponding Sylow p -subgroupof G hasa subgroupof order p i. The i i i product of these subgroups has order m, since G is a direct product of its Sylow subgroups. (cid:50) 7.8.6 Lemma (Frattini’s Argument). Let G be a finite group, and let H be a normalsubgroupofG. If P isanySylowsubgroupof H,thenG = H ·N(P),and [G : H]isadivisorof|N(P)|. Proof. Since H isnormalinG,itfollowsthattheproduct HN(P)isasubgroupof G. Ifg ∈ G,thengPg−1 ⊆ H since H isnormal,andthusgPg−1 isalsoaSylow subgroup of H. The second Sylow theorem (Theorem 7.7.4) implies that P and gPg−1 are conjugate in H, so there exists h ∈ H with h(gPg−1)h−1 = P. Thus hg ∈ N(P),andsog ∈ HN(P),whichshowsthatG = HN(P). It follows from the first isomorphism theorem (Theorem 7.1.1) that G/H ∼= N(P)/(N(P)∩ H),andso|G/H|isadivisorof|N(P)|. (cid:50) 7.8.7Proposition. Afinitegroupisnilpotentifandonlyifeverymaximalsubgroup isnormal. 488 CHAPTER7. STRUCTUREOFGROUPS(CONT’D) Proof. Assume that G is nilpotent, and H is a maximal subgroup of G. Then H isapropersubsetof N(H)byTheorem7.8.4andso N(H)mustequalG,showing that H isnormal. Conversely,supposethateverymaximalsubgroupofG isnormal,let P beany Sylow subgroup of G, and assume that P is not normal. Then N(P) is a proper subgroup of G, so it is contained in a maximal subgroup H, which is normal by assumption. Since P isaSylowsubgroupofG,itisaSylowsubgroupof H,sothe conditions of Lemma 7.8.6 hold, and G = HN(P). This is a contradiction, since N(P) ⊆ H. (cid:50) EXERCISES:SECTION7.8 1. ShowthatthegroupG isnilpotentifG/Z(G)isnilpotent. 2. Showthateachterm Z (G)intheascendingcentralseriesofagroupG isacharac- i teristicsubgroupofG. 3. Showthatanysubgroupofafinitenilpotentgroupisnilpotent. 4. (a)Provethat D issolvableforalln. n (b)Findnecessaryandsufficientconditionsonnsuchthat D isnilpotent. n 5. UseTheorem7.8.7toprovethatanyfactorgroupofafinitenilpotentgroupisagain nilpotent. 7.9 Semidirect Products Thedirectproductoftwogroupsdoesnotallowformuchcomplexityinthewayin which the groups are put together. For example, the direct product of two abelian groupsisagainabelian. Wenowgiveamoregeneralconstructionthatincludessome very useful and interesting examples. We recall that a group G is isomorphic to N×K,forsubgroupsN,K,provided(i)N andK arenormalinG;(ii)N∩K = {e}; and(iii) NK = G. (SeeTheorem7.1.3.) 7.9.1Definition. LetG beagroupwithsubgroups N and K suchthat (i) N isnormalinG; (ii) N ∩ K = {e};and (iii) NK = G. ThenG iscalledthesemidirectproductof N and K.