Abstract Algebra [Supplementary Lecture Notes]

25 Pages·2007·0.112 MB·English

by  John A. Beachy

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.