Annals of Mathematics

On Some Types of Topological Groups Author(s): Kenkichi Iwasawa Reviewed work(s): Source: Annals of Mathematics, Second Series, Vol. 50, No. 3 (Jul., 1949), pp. 507-558 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1969548 . Accessed: 31/01/2013 17:02 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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ANNALs OF MATHEMATICS

Vol. 50, No. 3, July, 1949

ON SOME TYPES OF TOPOLOGICAL GROUPS' KENKICHI IWASAWA (Received March 9, 1948)

Introduction As is well-knownthe so-calledfifthproblemof Hilberton continuousgroups was solved by J. v. Neumann [14]2forcompactgroupsand by L. Pontrjagin[15] forabelian groups. More recently,it is reported,C. Chevalley [6] solved it for solvable groups.3 Now it seems, as H. Freudenthal[7] clarifiedformaximallyalmost periodic groups,that the essential source of the proof of Hilbert's problemfor these groups lies in the fact that such groups can be approximatedby Lie groups. Here we say that a locallycompactgroupG can be approximatedby Lie groups, ifG containsa systemofnormalsubgroups{Na} such that G/Naare Lie groups and that the intersection ofall Na coincideswiththe identitye. For the brevity we call sucha groupa groupoftype(L) or an (L)-group. In thepresentpaper we shall study the structureof such (L)-groups,and apply the resultto solve the Hilbert'sproblemfora certainclass ofgroups,whichcontainsboth compactand solvablegroupsas special cases. We shall be able to characterizea Lie groupG, forwhichthefactorgroupGIN ofG moduloits radicalN is compact,completely by its structureas a topologicalgroup. The outlineofthepaperis as follows. In ?1 we studythetopologicalstructure of the group of automorphismsof a compact group and prove theoremsconcerningcompactnormalsubgroupsof a connectedtopologicalgroup,whichare to be used repeatedlyin succeedingsections. In ?2 come some preliminary considerationson solvable groups,whereas finerstructuraltheoremson these groups are, as special cases of (L)-groups,given later. In ?3 we prove some theoremson Lie groups. The theoremshere stated are not all new,but we give them here for the sake of completeness,and therebyrefineand modifythese theoremsso as to be applied appropriatelyin succeedingsections.4 Afterthese preparationswe study in ?4 the structureof (L)-groups. In particular,it is shown'that the studyof the local structureand the global topologicalstructure I In preparingthe presentpaper the author is greatlyindebtedto Prof. S. Iyanaga forhis kind encouragements and advices. The authorexpresseshere his best thanksto Prof.Iyanaga forhis kindness. 2 The numbers in bracketsreferto the bibliographyat the end of the paper. 3 Some of the books and papers as Chevalley [6],Malcev [12],whichwerepublishedin foreignlands duringthe last war,are knownto the authoronlyby the abstractsin Mathematical Reviews. Recentlyhe was also informedthat A. Malcev had provedin a paper of 1946,theoremson solvable groups,whichare similarto that for(L)-groupsin ?4 of the presentpaper. Howeverthe details of thesebooks and papers werenot attainableto the author. I This is also due to the circumstancesas stated in the footnote3.

507

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ofthesegroupscan be reducedto that of compactand Lie groups,groups,which we know, it may be said much about. The solutionof Hilbert's problemfor (L)-groupsthenfollowsimmediately. We also prove as a corollaryof a general theoremthat any connectedlocally compact solvable group is an (L)-group, so that our result covers Chevalley's theoremreferredto above. In the last section,?5, we study the generallocally compact groupsin makinguse of the results obtained in preceding sections. Particularly it is proved that any connected locally compact group has a radical, i.e. a uniquely determined maximal connectedsolvable normal subgroup. By this notion of the radical we definea special typeof topologicalgroups. We say namelythat a connected locally compactgroupG is a (C)-group,when the factorgroupG/N modulo its radicalN is compact. It is thenprovedthat any (C)-groupis an (L)-group,so that Hilbert'sproblemis also solved for(C)-groups. This containsboth results of v. Neumann and of Chevalley. Afterthese considerationsthere remainsa question:what is the situationof the class of (L)-groupsin the set of all locally compact groups? It seems likely to us that any connected locally compact groupis an (L)-group. In connectionwiththis conjecturewe also show that it is equivalent to a slightlystrengthenedhypothesisof C. Chevalley.5If this conjectureturnsout to be valid, resultsin ?4 would give us the structureof generalconnectedlocally compact groupsand the Hilbert's problemwould be thencompletelysolved. Althoughour investigationsare not completein various view-points,we hope that the presentpaper makes a contributionto the foundationof the theoryof locallycompactgroups. 1. On the groupof automorphismsof a compactgroup Let K be a compact topologicalgroup and A (K) the set of all continuous of K. A (K) formsan abstractgroup; we shall introducein the automorphisms followinga topologyin A (K), so that it becomesa topologicalgroup. For that oftheidentityin K and put purposetake any V in a systemI VI ofneighborhoods (1.1)

U(V) = {a; a e A (K), s?'- e V, forall s in K} .

If we take the familyI U(V) } as a systemof neighborhoodsof the identityin A (K), it is easy to see that A (K) becomesa topologicalgroup. In the following we always considerA (K) withsuch a topology. The same topologycan be introducedalso as follows. Let C(K) be the set of all continuous(complex-valued)functionson K and defineforany f(s) in C(K) and a in A (K) the functionfe(s) by f?(s) = f(sU) Obviouslyf' is also containedin C(K), and we have 6 Cf. H. Cartan [4].

IIf?11= llf11,

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SOME

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OF TOPOLOGICAL

509

wherethe norm II f jj is definedby

IIf1f = MaxIf(s) ,

Now take any fi, *

s inK.

in C(K) and e > 0, and put , fA

...

,n}. e) la;={;a e A(K), jfs -fi 11 < ei = 1 Xf ; e) } also definesa systemof neighborhoodsof the X The family { U(f1,

U

identityin A (K) and we can see immediatelythat this topologyis equivalent to the one definedabove by { U(V) }. It is herealso to be noted that,ifK is connected(1.1) can be replaced by the family (1.2)

U'(V)

=

{a; a

e

A(K), s'-' e V, forall s in Vol,

whereVo is an arbitraryfixedneighborhoodof the identityin K. Now let G be a topologicalgroup,which containsK as a normal subgroup. Any g in G thendefinesan automorphisma(g) in K by (g) =g sg,

seK.

It is theneasy to see that the homomorphicmapping (1.3)

g -9 a(g)

fromG intoA (K) is continuousin the sense of above definedtopologyin A (K). If we take in particularG = K, we see that the imageI(K) ofthe mapping(1.3), i.e. the group of innerautomorphismsof K is a compact normal subgroup of A (K). We can thereforeconsiderA (K)/I(K) also as a topologicalgroup. Now themainpurposeofthe presentsectionis to provethefollowingtheorem. 1. A (K)/I(K) is a totallydisconnected (0-dimensional)group. THEOREM We firstprove a special case of the theorem,namely LEMMA 1.1. If K is a compactLie group,A (K)/I(K) is discrete. We shall reduce the proofstep by step to the case whereK has a PROOF. moresimplestructure. i) Let Ko be the connectedcomponentof the identityin K. K/Ko is then a finitegroupof order,say, n. We prove firstthat if the lemma is valid forKo, it is also valid forK. Denote by Al the set of all automorphismsof K which leave invarianteverycoset of K/Ko, and by I*(Ko) the set of innerautomorphismsofK inducedbyelementsin Ko . Al is a closed normalsubgroupofA (K) with a finiteindex, for A (K)/A1 is ismorphicto a subgroupof the group of automorphismsof finiteK/Ko. Al is consequentlyopen in A(K). I*(Ko) is also a normalsubgroupofA (K) and is containedin I(K). ThereforeA (K)/I(K) will be surelydiscreteifwe can provethat A /I*(Ko) is discrete. subgroupofK, any a in Al inducesan automorphism As Ko is a characteristic a' in Ko and the homomorphicmapping a -a

= (p(a)

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IWASAWA

fromA1 intoA (Ko) is continuous. Let N be the kernelof this homomorphism. Now fromthe definitionof A1, we have fora in A1 s8 = su,

se K,

u e Ko,

whereu = u(s) is a continuousfunctionof s. If in particulara is containedin N, u(s) is an elementof the centerZ of Ko and it dependsonlyupon the coset of K/Ko containingS.6 From (ss')Y = s~s's we have then u(ss')

(1.4) If we put

II

= u(s) u(s').

(s'u(s)s')u(s')

=

u(s) = v,

s mod. Ko,

= u(s')n,

forall s' in K.

it followsfrom(1.4) that v

(1.5)

As Z is a compactabelian (not necessarilyconnected)Lie group,we can finda finitecharacteristicsubgroupZ1 of Z, so that Z/Z1 is connected,i.e. a toroidal groupof some finitedimension. Thereforethereis an elementw in Z such that wn

mod. Z,

_v-

If we denoteby a, the innerautomorphisminducedby the elementw in K and ifwe put a2

=

alaX

=

s e K,

su'()

u'(s)e

Z.

a simplecalculationshowsthat we have (U, (s)) ne Z

in virtueof (1.5). Let Z2 be the set of all elementsin Z, whose nth powerfall in Z1 . Z2 is then also a finitegroupand u'(s) can be consideredas a function fromthe finitegroupK/Ko into finiteZ2 . As thereexistsonly a finitenumber of such functions,it followsthat the index [N:N

I*(Ko)]

is finite. On the otherhand we have obviously and it followsthat the index

p(I*(Ko))

= I(Ko)

[ [hi, h2], hi e H1 fromH1 into G. As the image [H1, h2]of this mappingis a connectedsubset containinge = [e, h2],it follows

[H1,h2]C Mi. Since h2is an arbitraryelementin H2, we have then [H1, H2] C M, whence C(H1, H2) C MI,C(H1, H2) = M, q.e.d. subgroups{Dt(G)} of G Now we defineby inductionthe seriesof commutator as follows. First we put Do(G) = G and suppose that D,(G) are already definedfor all ordinal numbers tq < t. We put then D(G) = C(Dtw(G),Dt'(G)), if t = i' + 1 and

D(G) = n,f aua-1,

ue N

of N, and it can be seen that G/M is isomorphicto a compact group of linear in a vector space V = N. G/M is consequentlya compact transformations N is containedin the centerof M. Lie group. Now by the definitionof M11, As N contains two discrete subgroupsD1, D2, such that N/D1, N/D2 are = e, we see that M is a maximallyalmostperiodicgroup. compactand Di D Moreoverthe factorgroupM/Mo of M modulo its connectedcomponentMo is compact,forMo obviouslycontainsN. M is consequentlythe directproduct of N and a characteristiccompact subgroupK1.17 Consider then G/K1and apply Lemma 3.7 with respectto its normalsubgroupMI/K _-N. It follows thenimmediatelythat thereexistsa compactsubgroupK such that G=KN,

K

N=e.

The second part of the lemma, i.e. the conjugacy of such K holds also in the part of the proofof Lemma 3.7 has not used generalcase, forthe corresponding the fact that G/N is a Lie group. topologicalgroupand N a normalsubgroup LEMMA3.9. Let G be a connected Lie group,G containsa subgroupH, such semi-simple of G. If N is a connected as HN = G and H , N is a discrete G/N is a simplyconnected group. If moreover Lie group,we may takesuchH as H N = e. PROOF. If we take a system of canonical coordinatesof the first kind of N inducesa lineartransformation in {. ) * * ,X,, in N, any automorphism In particularthe automorphisms u->aua-, ueN, inducedby elementsa in G, can be representedas a connectedlinear group on {i . Now it is well-knownthat the connectedcomponentof the groupof automorphismsof a semi-simpleLie group just coincideswith the group of inner automorphisms. If we denote thereforeby H the set of elementsin G, which commutewith every element in N, we have, as in the proof of Theorem 2, G = HN. The groupH , N is discrete,forit is the centerof semi-simpleN. Suppose next that G/N is a simplyconnectedLie groupand let Ho be the connected componentof H. It is then easy to see that we have G = HoN. It followsthen G/N - Ho/Ho N 17

Cf. Freudenthal[7].

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525

SOME TYPES OF TOPOLOGICAL GROUPS

and this shows that Ho is locally isomorphicto simplyconnectedGIN. As Ho is connected,it followsthenHo N = e, as asserted. LEMMA3.10. Let G be a locallycompactgroupand N a normalsubgroupof G. If N is a connected toV1 , G containsa subgroup Lie groupand if GIN is isomorphic H suchthatG = HN, H N = e. PROOF. Let N1be the radical of N. Applyingthe previouslemma to GIN, and N/N1, we see that thereexists a subgroupH1 in G such that G = H1N, H1 -. N = N1, H1/N1_ V1. By Lemma 3.4 thereis then a subgroupH so thatH1 = HN1, H - N1= e. It is theneasy to see thatG = HN, H - N = e. LEMMA3.11. Let G be theadjoint groupof a real semi-simpleLie algebraL. Thenthereexistsa connected, simplyconnected solvablesubgroupH and a maximal compactsubgroupK of G, so that G = HK, K = e, H -

i.e. thatany elementg in G can be written uniquelyin theform h eH, keK

g = hk,

and h and k dependcontinuously on g. In particularthespace ofG is theCartesian is homeomorphic toa Euclidean productofthespacesofH and K, ofwhichtheformer space. PROOF.18 Let L be the complexformof the semi-simpleLie algebra L of degrees and Lo the compact formof L, i.e. the Lie algebra which corresponds to a compact semi-simpleLie group and whose complex form is L. By a theoremof E. Cartan19the algebra L, or more rigorouslyspeaking,an algebra whichis isomorphicto L can be obtainedfromLo in the followingway. Take a suitableinvolutiveautomorphisma of Lo and choose a real basis of Lo 81, *@,us

such that

a(ui) =

fori = 1, - r

ui,

fori = r + 1,*** ,s,

= -U,,

forsome integerr. A real basis of L can be then obtainedby putting Vi =

ui

r =

*.*.*.

Ur

V/-lur+i

Vr

***v=

-/iU

.

In otherwords,ifwe put u = u' + u",

(3.6)

u'

=

2-(U +

cr(u') = u',

a(u)), a(u")

u" =

=

(U-a(u)),

-u",

forany u in Lo, then 18 The author's originalproofwas only valid for complex semi-simple Lie algebras. The presentproofis due to Prof.C. Chevalley,to whomthe authorwishesto expresshis heartythanks. 19Cf. E. Cartan [3] or Gantmacher[8], [9].

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526

KENKICHI

(3.7)

L

=

IWASAWA

u' + v

=

_utt}

and consequently Lo = L

(3.8)

(u') = u'}.

{u'; u' e Lo,

Lo=

In the followingwe denote also by a the automorphismof L,whichis induced by a on the complexformL. Now we take the maximum number of mutually commutative linearly independentelementsh', ** , h. in Lo such that a(hs) = -hi'

i=1,

,m.

Let Ho be a maximalabelian subalgebraof Lo containingthese h'. If h is an arbitraryelementin Ho, h, a(h) and consequentlyh' = h - a(h) are all cQmmutativewithh' (i = 1, *..* m) and we have moreovera(h') = -h'. According to the choice of h', * , A , h' must be contained in the linear set } and this shows that a(h) e Ho and Ho is invariantby the auto.h **, morphisma. AfterWeyl20we can then choose a complexbasis .

hl, * * hnX ea X e-a

X .. *

ep ,

e-,

of L as follows: [hi, hi] = 0,

[Ei::~-l v Xea] = (3.9)

[eaXea]

(aX) =

(aX)ea,

=-Viol, aeiha

2z:LiatiiX

, [ea) e#]= Na,#ea+#

(a

0).

+B3

Here a = (al, ... , an) are real vectors,called root vectorsof L and Na, 7? 0 if and only if a + ,3 is again a rootvector. MoreoverLo can be identifiedwith the set of elements X=

x/-lXihi +

Taea,

Ea

Xi = real,

T-a

=

Ta

and in particular

Ho = {h; h =

Zn1

NviZXjhi Xi = real}.

We may also suppose that {h', (3.10) (3.10)

oQVaihi)

, = =

h

=

-Vphi

vzhi,

{V/-lhl,

,

If we thereforeconsider(X1, , X,) as coordinatesof h i.. a inducesa lineartransformation 20

Cf. Weyl [18].

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*,

i

=

V/-1hm}and 1, ***,mM

i=m+1, =

,n.

V/-1Xihi,

TYPES

SOME (3.11)

.((X,

=

Xn))

X

(-

X

***

527

GROUPS

OF TOPOLOGICAL -Xm,

Xmn+l

X

*

I Xn)

in the X-space. In particularforany root vector a of L we have from(3.9)

[V(Es1 v'kihi), [-Et

-1 V"ihi

= (aX)o(e,),

c(e,)]

+

Et

m-+ /ihi,

o(e)] = (u(a)u(X))u(e),

whichshowsthat a' = v(a) is again a rootvectorofL and that a(e,) is an eigenelementbelongingto a'. In orderto see the relationbetweena and a' = a(a), we definea linearorder in the above X-space,so that it becomes a linearlyorderedabelian group. We put namely (X

X* *X

n)>

n)

(X

ifand onlyifx *i, , * *, Xk-i= Xki-, Xi> X forsomei. It thenfollowsimmediatelythat for any root vector a > 0, we have eitherox(a) < 0 or v(a) = a. We denote by E the set of all a > 0 with v(a) < 0 and by E' that of a > 0 a. For any a e E' it holds then with o(a) (3.12)

o(ea)

= ea.,

o(e-a)

= en

For v(a) = a means by (3.11) that the firstm coordinatesof a vanish and this h' * * , I implies that e, and ea are both commutative with every element in Ihi, = the elements Therefore (e, + e,) V"-hm {i/Ihl, hi *, u, = of and are also e-a) u2 x/1111(ea + e-a) e-,)) Lo o(e, 07(V'ZT(e. commutative with ha (i = 1, * , m) and satisfy a(ul) = -ui, a (u2) = u2 By the choice of hi it follows u, = u2 = 0, a(eC) = ea, o(e-a) = e_, as stated

above. Now let R be the set of all elements

+ ZEoE ,Te., =Et- 1KAhi (3.13) where Ki and ra run througharbitrarycomplex numbers. If we note that > a, f, we can see easilythatR is a solvable a, A e E impliesa + A e E and a + = L L. R' of R, we shall prove then Putting subalgebra L (3.14) R'+L'0 to prove L C R + L' whereL' was given by (3.8). It is of course sufficient or by (3.6), (3.7), (3.8) to show that V (u - a(u)) is containedin R + Lo forany u in Lo. As i = 1, ***, ~~n,

v'hhi, e,

+ e.a ,

x/-l(e, - e,),

a > 0

forma real basis of Lo, we have only to prove that the elements =n1,

,i

(v'Hhi-~

,

X/-1i((ea + e-a) -o(ea + e-a)),

V'/ZII(V/ZII(ea - e-)

-o(V

i(ea

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-

ea))),

a > 0

528

IWASAWA

KENKICHI

are containedin R + Lo'. Again by (3.10), (3.12) it sufficesto show that N/fZ1((e. + e-a)

+ ea.)

-

=

a(ea + e-a)) -

1,*2*

,

m) this is obvious. For the (e,))

-1(ea-,

-(V-1(e.

-

e-,) +

,m

a eE

(ea-ea)-o(ea-ea),

(ea + e-a)),

-

are elementsof R + Lo'. As to hi (i otherelementswe have (e,

**

i-1,

hi,

o(V-IZ(ea

mod. R

- ea)))

+ e-a) -a (ea + e-a)) and thelast side is obviouslyan elementofLo. V-1((ea containedin R + L' and so is (e, - ea)o-(ea,- ea) by a similar is therefore reason. We have thus proved (3.14). Now let G be the adjoint group of L. The adjoint group G of L is then a H and K be connectedclosed subgroupsof G, closedsubgroupofG. Let further to the Lie algebras whichare generatedby local subgroupsof G, corresponding R' and Lo respectively. As K is clearlycontainedin the compactadjoint group of Lo, it is also a compactsubgroupof G. On the otherhand the coefficients Ki in (3.13) of any elementy in R' are, as one readilysees, all real and the matrix ofL withrespectto the basis fory in the regularrepresentation hi,-

has the form

,hn ye.

:,eft ,.ep;

a



H'H'/Z. K' = e.

Here K' is a connectedgroup,whichis locally isomorphicto the compactgroup K'/Z. It is thereforeeitheritselfcompact or the directproductof a compact groupK and a vector group V. Moreoverit is easy to see that K' or K is a maximalcompact subgroupof G. Next let K* be an compact subgroupof G. K*Z/Z is thenalso a compactsubgroupof G/Z and by a theoremof E. Cartan21 some conjugate of K*Z/Z is containedin K'/Z. A conjugate subgroupof K* must be then containedin K' and even in K, if K' is not compact. Thus it can be seen that maximalcompactsubgroupsof G are conjugateto each other. By puttingHo' = H, K' -M, i), ii), iii) are proved and the assertionon the topologicalstructurefollowsalso immediately. THEOREM 6.22 Let G be an arbitraryconnectedLie group. Then maximal and conjugateto each other. Let K be one of compactsubgroupsofG are connected to V1 them. ThereexistthensubgroupsH1, - **, Hr ofG, whichare all isomorphic and are suchthatany elementg in G can be decomposeduniquelyand continuously in theform (3.17)

g = hi h.lt,

hi e Hi,

k e KK,

or briefly expressed: G = H1i.

HK.

In particularthespace of G is theCartesianproductof thecompactspace ofK and Euclideanspace Er . tother-dimensional thatofH1 ... Hr, whichis homeomorphic if PROOF. If G is a simpleabelian group,namely G is isomorphicto V1 or T , the theoremis evident. If G is simple and semi-simple,the theoremfollows immediatelyfromLemma 3.6 and Lemma 3.12. Thereforeif we apply the inductionwithrespectto the dimensionof G, the proofforthe generalcase will be completedby the followinglemma,whichwe shall prove in a slightlymore generalizedformthan is requiredfor the presentproof,for the sake of later applications. LEMMA3.13. Let G be a locallycompactgroupand N a normalsubgroupof G. Lie groupsand if Theorem6 is alreadyproved If N and GIN are bothconnected are also validforG, evenwhen same the propositions then to betrite forthesegroups, 21 Cf. E. Cartan [31. 22 Cf. Malcev [121.

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531

we do notassumethatG is a Lie group. Moreoverif K1 is a subgroupofG containingN suchthatK1/N is a maximalcompactsubgroupofG/N and ifK2is a maximal compactsubgroupof N, we can finda maximalcompactsubgroupK of G, which satisfies K1 = KN,

(3.18)

K2=

K

N.

PROOF. Suppose firstthat N is semi-simple. By Lemma 3.12 N has a

subgroupM = K2 X V, in whichthe centerofN is contained. Now by Lemma 3.9 the set of all elementsin G, whichcommutewitheveryelementin N, forma N is the discrete normalsubgroupH, forwhichwe have G = HN and Z = H Z V. By a = M we Z1 a subgroupof and centerof N. Z is therefore put a group. suitablechoiceof V we may suppose that Z/Z1 is finite Now as is readilyseen,H has a subgroupK3 containingZ, such that Z =K3

K1-=KN,

K31/Z_ K1/N.

N

As K3/Z is compactand Z/Z1 is finite,K3/Z1 is also a compactgroup. CombinV = Z, we see that K3V is a closed subgroup,V is normalin ing thiswithK3 that K3V/V is compact. By Lemma 3.8 thereexists consequentlya K3V and compactgroupK4 such that K3V = K4V,

K4

-

V

=

e.

Now the elementsof K2 are commutativewith elementsof both K3 and V, a compactsubgroupofG. consequentlywiththat ofK4. K = K2K4is therefore N. On the otherhand KN As it containsK2 we see immediatelythat K2 = K containsK3, consequentlyK1. As K1/N is a maximal compact subgroupof GIN, it followsK1 = KN. The existenceof a compact group K as (3.18) is thus provedforthe semi-simplegroupN. Now under the same assumptionforN, let K* be any compact subgroupof G. As K*N/N is a compact subgroup of GIN, some conjugate of K*N/N mustbe containedin K1/N by the assumptionofGIN. We can findaccordingly in K,. We may suppose therefore an elementin G, by whichK* is transformed from the beginningthat K* is contained in K1. Now consider K*Z/Z in G/Z = H/Z X N/Z. It is theneasy to see thatK*Z/Z lies in thedirectproduct ofK3/Z and a maximalcompactsubgroupofN/Z. As N/Z is the adjoint group of the semi-simpleLie group N and as one of maximal compact subgroupsof N/Z is given by M/Z,23we may suppose, if necessaryaftera suitable transformation,that the latter directfactoris just M/Z. K* is consequentlycontained in K3M. On the otherhand this groupis equal to K3VK2 = VK4K2 = VK = V X K. The compact group K* must be thereforecontained in K. This proves that K is a maximal compact subgroup of G and that all such groupsare conjugateto each other. To provethe like propositionson K forthe generalcase, we use the induction 23

See the proofof Lemma 3.12.

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532

KENKICHI

IWASAWA

with respect to the dimensionof N. By the above considerationswe may suppose that N is not semi-simple. By Lemma 3.3 N containsthen a normal subgroupN1 of G, whichis isomorphiceitherto a vectorgroup or to a toroidal group. Let us considerfirstthe formercase. By makinguse of Lemma 3.7, we can see readilythat K2N1/N1is then a maximal compact subgroupof N/N1. By the inductionassumptionapplied to GIN1 and N/N1, there exists then a subgroupK3 ofG containingN1 , suchthatK3/N1is a maximalcompactsubgroup of GIN1 and that K1 = K3N, K2N1 = Ka N. By Lemma 3.8 K3 containsa compactsubgroupK, such as K3

=

K , N1 = e.

KN2,

As K2 is a compactsubgroupof K3, we may suppose moreover,if necessaryby takinga conjugategroup,that K containsK2. We can see then immediately that such K satisfiesthe relations(3.18). Now let K* be an arbitrarycompact subgroupof G. As K*N1/N1is a compact subgroupof GIN1, some conjugate of it must be containedin K3/N1. Consequentlysome conjugate of K* lies as one readily sees by in K3 and even in K after a suitable transformation, Lemma 3.8. This completesthe proofforthe firstcase. Next we considerthe case, whereN, is a toroidalgroup. In this case N, is containedin K2, forthisis a maximalcompactsubgroupofN. By thedefinition N, is also a subgroupof K1. Thereforeall considerationscan be easily reduced to that of the factorgroup G/N1, forwhichthe propositionsare proved to be valid by the inductionassumption. Our assertionson the maximal conjugate subgroupsof G are thus completelyproved. Now let H', ** , H' be subgroupsof G containingN, such that GIN is the productof H'/N and K1/N as stated in Theorem6, and let Hs+l, , Hr be HrK2. By Lemma 3.10 thereexist subgroupsof N, such that N = HL4Hs in G, so that subgroupsH1,i .

.

H' = HiN,

Hi

N = e,

i =1,---,s.

It is then easy to see that any g in G can be decomposeduniquelyin the form (3.19)

g = hi ..* hki ,

hi e Hi , ki e K,

and thathi and k1depend continuouslyon g. Now we have alreadyprovedthat K1 = KN, K2 = K N, whence followsK1 = NK = H8+1 ... HrK2K = H8+i ... HrK. Any ki in K1 can be writtenconsequentlyin the form (3.20)

ki = h8+i**hrk,

hieHi,

keK.

The uniqueness of this decompositionis obvious. If a sequence k") in K1 convergesto k1, we can choose a subsequence so that its K-componentsV(j) convergeto some element k in K. Then the remainingH-componentsalso

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SOME

TYPES

OF TOPOLOGICAL

GROUPS

533

convergeto someh. Howeveras the set H8+*... Ho is closedby the assumption, h must belong to H,+1 ... Hr. Thereforek1 = hk is the decompositionof k1 intoh in H1141***Hrand k in K. This showsthe continuityofk; that ofeach hi followsfromthe assumptionon N. Combining(3.20) with (3.19), Lemma 3.13 is thus completelyproved. Now we shall give some consequencesof Theorem6 and Lemma 3.13. LEMMA3.14. A connectedLie groupG is simplyconnected, if and only if a maximalcompactsubgroupK of G is simplyconnected. PROOF. It followsat once fromTheorem6. DEFINITION. We call theintegerr in Theorem6, namelythedimensionof the cosetspace of a connected Lie groupG withrespectto a maximalcompactsubgroup K, thecharacteristic indexof G. LEMMA3.15. Let G bea connected Lie groupand N a connected normalsubgroup N and KN/N are maximal ofG. If K is a maximalcompactsubgroupofG, K compactsubgroupsofN and G/N respectively and thecharacteristic indexofG is the sum of thecharacteristic indicesofN and G/N. MoreoverG is simplyconnected, if and onlyif N and G/N are bothsimplyconnected. PROOF. By Lemma 3.13 thereexistsa suitable maximal compact subgroup K of G, so that K N and KNIN are maximalcompact subgroupsof N and GIN respectively. However the same is also true for any other maximal compact subgroupof G, for these groups are conjugate to each other. The assertionon the characteristicindex followsalso immediatelyfromthe considerationsin the proofof Lemma 3.13. On the otherhand it is easy to see by a structuraltheoremon compactLie groups,24 that the compactgroupK is simply connectedifand onlyifK - N and K/K N are both simplyconnected. As KN/N _ K/K - N the last assertionthenfollowsfromLemma 3.14. REMARK. Even when we do not assume that G is a Lie group, but only assume that N and GIN are both connectedLie groups,all above propositions hold as well. Note that we can prove in such a case that K is a compact Lie group, for it can be readily seen that K does not contain arbitrarysmall subgroups. LEMMA3.16. Let G be a locallycompactgroupand N a normalsubgroupof G. Let further N and GIN be bothconnected Lie groups. Then thesimplyconnected normalsubgroupIV, such thatX and coveringgroupG of G containsa connected and thattheyare locallyisomorphicto N and GIN O/A are bothsimplyconnected respectively. PROOF. it can be seen that G has a neighborhood By the above considerations of e, whichis homeomorphicto a Euclidean space. Thereforewe can construct reallythe simplyconnectedcoveringgroupG over G, whichis locallyisomorphic to G. Let spbe thehomomorphic mappingfromG onto G and let us denoteby At the connected componentof -'(N). Then N, f'(N) and iV are all locally isomorphicand the same is also true for GIN and O/iV. At and O/2S are 24

See forexamplePontrjagin[16],?55.

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534

IWASAWA

KENKICHI

consequentlyboth connected Lie groups. That they are moreover simply connectedfollowsfromLemma 3.15 and the remarkafterit. Now we shall give some lemmas,whichwill be used in the proofof the next theorem. locallycompactgroupG containsa normalsubgroup LEMMA3.17. If a connected G = HN, Lie groupsand if moreover N and a subgroupH, whichare bothconnected thenG itselfis a Lie group. Let M = H , N and take a systemof one-parametergroups PROOF. such that

H -Li ... L,,

1,

{xi(Xi); I Xi I < e}l-

Li No-

t,

Lr+,... L x M--L,+j ... Ls,(r < s < t),

whereH - L1 ... L8 means forexamplethat any h in H, whichis sufficiently near to e, can be expresseduniquelyin the productform h

=

xi(Xi) ...

x8(XA)

{

Xi I < E.

It is theneasy to see that any elementg in some neighborhoodof e in G can be uniquelydecomposedinto the product 9 = xi(X%)

(3.21)

. .

xr(Xr)xri(Xr+)

. . .

. . .

x,(X,)

xi I