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A GENERALIZATIONOF THE CARTAN-BRAUER-HUA
THEOREM
211
2. G. Higman, On finite groups of exponent five, Proc. Cambridge Philos. Soc. 52
(1956), 381-390.
3. A. I. Kostrikin, On Burnside's problem, Dokl. Akad. Nauk SSSR 119 (1958),
1081-1084. (Russian)
4. M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. École Norm.
Sup. (3) 71 (1954), 101-190.
5. H. Zassenhaus,
Ein Verfahren, jeder endlichen p-Gruppe eine Lie-Ring mit der
Charakteristik p zuzuordnen, Abh. Math. Sem. Hamburg Univ. 13 (1939), 200-207.
University
of Wisconsin
A GENERALIZATION OF THE CARTANBRAUER-HUATHEOREM
CHARLES J. STUTH
Let A be a division ring. Then K' will denote the multiplicative
group of K. If 5 is a subset of K, then S* will denote the division
ring generated by S, and C(S) will denote the centralizer of 5 in K.
If G is a subgroup or a subdivision ring contained in K, then Z(G)
is the center of G. If x and y are elements of K', then (x, y)=xyx~1y~l
and xv = yxy~l. If b is an element of a group A, then Cl(A, b) will denote the group which is generated by the conjugate class of b in A.
A set is central in K if each of its elements is in Z(K). A group G
is «-subnormal
in a group H il there are groups G\, ■ • • , Gn-x such
that G = G„<]Gn_i<] • • • <lGi<]Go = .r7. A group G is subnormal in a
group H if G is an «-subnormal
subgroup of H for some n. A group G
is invariant under a group H if ghEG for all gEG, all hEH. A noncentral subgroup G is of type I if G is invariant under a noncentral
subnormal subgroup of K'. K has the property P„ if for every subdivision ring H of K such that H' is invariant under a noncentral nsubnormal subgroup of K' it follows that H is central or H = K.
The Cartan-Brauer-Hua
Theorem [2] states that a division ring
has property
P0. Herstein and Scott [l] generalized this to Pi.
Schenkman
and Scott [5] extended the Cartan-Brauer-Hua
Theorem by showing that a division ring has property P„ for all n if each
of its subdivision rings which is invariant under a subnormal subgroup is normal in some subnormal subgroup of the division ring.
Theorem 1 of this paper shows that a division ring has property
P„ for all n. Then results are developed from this concerning the
subnormal subgroups of K' and more generally for the subgroups of
type I in K'.
Received by the editors January 16, 1963.
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212
C. J. STUTH
Lemma 0. Let xEK,
iy, l+x)=yi,
yi=iy,
[April
yEK, (y, x) ^ 1 commute with both x and y. Let
y<-i). If some yn=l, then x is algebraic over
ZiK).
Proof.
This is Lemma
1 of [l] and follows immediately
from
Lemmas 1 and 2 of [4].
Theorem 0. Suppose every division ring has property P„_i. Then if
H is a subfield of K invariant under G„<] ■ • ■ <$Gi<\Go = K', where
Gn is a noncentral subgroup of K', then HEZiK).
Proof. This is Theorem 2 of [l].
Lemma
1. Let Go, • • • , Gm be groups
such that G¡ <¡ Gy_i for
j=l, • ■ • , m with hEGm. Furthermore let G(0, 0) = G0 and G(i, 0)
= Cl(G(i —1, 0), h) for i=l, • • ■,m. Then G(m, 0) is m-subnormal in
Gm.
Proof. For any group A with xEA, C\(A, x) is the smallest subgroup containing x which is normal in A. So if B is a normal subgroup
of A with xEB, then C\(A, x)<\B.
Let G(i, i) = d for i=l, ■ ■ ■, m. Assume inductively for all i<k
that G(* - 1, 0)< ■• • < Gii - 1, • - 1), where G(i - l,j - l) =
Cl(G(i-2,j~l),h)iorj=l,
■■■,i-l.LetGik,j)
for j = 0, ■ ■ • , k —l. By the remark
G(k, j)<\G(k-l,
j-l)
for j=l,
= C\iGik-l,j),h)
at the beginning
■■■, k-l.
of this proof
By hypothesis G(k, k)
<\G(k —l, k —l). Then, again by the remark above, G(k, j —l)
<Gik,j)iorj=l,
■ ■ ■,k.
Lemma 2. Let G be a group with h EG. Let M0 = G, Mj = C\(Mj-i, h)
for j—l, ■ ■ • , »t. Then M¡ is invariant
where H is the centralizer of h in G.
under H for j = 0, • • • , m
Proof. It is seen that Mm<\ • ■ ■ <\M0. Assume inductively
that
Afi_i is invariant under H for some i — Km.
Let y EH, gEMi-i.
Then h'EMi
and g"G.M<-i so that (hä)y= g!'h(g-l)''EMi. Thus,
xvEMi for all xEMi. Hence M i is invariant under H.
Until Theorem 1 suppose that every division ring has property
P„_i, and that K is a division ring with a proper subdivision ring H
which is not central in K and is invariant under G„<] • ■ ■ <\Gi<\Go
= K', where G„ is a noncentral subgroup of K'.
Lemma 3. Let G be a noncentral n-subnormal subgroup of K'. For
each noncentral kEK there exists gEG such that (g, k) is not central in K.
Proof.
Suppose there is some noncentral
kEK
such that (g, k) is
central for each gEG. Then k"= (g, k)k so that C(k") = C(k) for each
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1964I
A GENERALIZATIONOF THE CARTAN-BRAUER-HUA
THEOREM
213
gEG. Thus k and its G-conjugates
commute, and they generate
a noncentral subfield F which is invariant under G, contrary to
Theorem 0.
Lemma 4. Let M be a noncentral subgroup of K' which is invariant
under G„. Then MC\Gn is not central in K.
Proof. Assume inductively that there exists noncentral hEMC\Gi-x
for some i—l <n. By Lemma 3 there exists gEGn such that (h, g) is
not central. Since (h, ¿)EMC\Gi, then M(~\Gi is not central.
Lemma 5. Let h be a noncentral element in Hr\Gn.
gEGn\H
Then there exists
such that (g, h) is not central.
Proof.
If GnEH, then by P„_i, K = (Gn)*EH, contrary to sup-
position on H. Therefore
Gn($_H, and there exists xEGn\H.
Suppose (g, h) is central for all gEGn\H. Let wEHi\Gn. Then
wxEGn\H. (wx, h)=w(x, h)hw~lh~1=(x, h)(w, h). Since (wx, h) and
(x, h) are central, then (w, h) is central.
all gEGn, contrary to Lemma 3.
Therefore
(g, h) is central for
Lemma 6.
(i) C(a)EH for all aE(HC\Gn)\Z(K) ;
(ii) C(H)=Z(K) = Z(H).
Proof.
Since H is invariant
under G„, then Z(H) is a field invariant
under G„. By Theorem 0, Z(H) EZ(K). If (i) is true, then C(H) EH,
so that Z(K)EC(H)=Z(H)
QZ(K). Hence it suffices to prove (i).
By Lemma 4 there exists hEHC\Gn
C(h) (t H. Then
Mi=Cl(Mi-x,
in K'.
cm-
there
which is not central.
exists y E C(h)\H.
Let
h) lor i= 1, • • • , n. M„ is noncentral
By Lemma
1, MnEGn.
By Lemma
Suppose
M0 = K'
and
and «-subnormal
2, Mn is invariant
under
Let gEMn. Then h"EH and gyEMn. Hence gvh(g-i)v = yghg-ly~l
= c is in H. Let z= 1 —y. Since zEC(h)\H, then zghg~lz~~l= d is in H.
Therefore yh" = cy and zh" = dz. Adding, it follows that h° — d
= (c —d)y. If c —d is not zero, then yEH,
a contradiction.
Hence
d —c = 0 = h' —d, so that h° = c. Therefore yEC(h"). If w is an element
of C(h)C\H, then both y and y-w
are in C(h)\HEC(h°),
so that
wECQi").Hence C(h)EC(h°) for all gG-M».
If bECih) and ^r'Élí,,
then bEC(hl), and thus b*EC(h).
Therefore C(h) is invariant under ¥». Then Z(C(h)) is invariant
under Mn and is a noncentral field in K because hEZ(C(h)),
con-
trary to Theorem 0. Hence C(h) EH.
Remark. We now let Z = Z(H)=Z(K)
until Theorem 2.
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214
C. J. STUTH
[April
Lemma 7. Let g be an element of the normalizer of H in K, gEH.
Then Hr\Hk=C(g)C\H,
Proof.
where k = 1 +g.
If xEC(g)C\H,
then x = xk, so xEHC\Hk.
Hence C(g)C\H
EHC^BT.
Let hEH(~\Hk. There exists jEH such that h=jk. By hypothesis
there exists mEH such that h = m°. Then hk = kj and hg = gm. Subtracting, it follows that h—j = g(j —m). If j —m is not zero, then
gEH, a contradiction.
Soj —m = 0 = h—j and h = m. Then h = h° so
that ÄGC(g). Therefore HC\HkEC(g)C\H.
Corollary.
Under the hypothesis of Lemma 7, GnP\HC\HkEZ.
Proof.
By Lemma 7, Gn H H C\ Hk = Gn C\ H H C(g). If
hE (GnC\HC\C(g))\Z, then gG C(ä) EH by Lemma 6, a contradiction.
Lemma 8. There is no element in (HC\Gn)\Z
over Z.
which is algebraic
Proof. Suppose hE(HC\Gn)\Z
is algebraic over Z. By Lemma 5
there exists gEGn\H. Let Z(h) be the field generated by adjoining
h to Z. Now h and h" have the same minimal equation so there is an
isomorphism between Z(h) and Z(h") (induced by h+-*hs) leaving Z
elementwise invariant.
By Corollary 2, p. 162 of [3], there exists
xEH such that x induces the same inner automorphism
as g does.
Thus, ha = hx from which x~lgEC(h).
x~lgEH.
Therefore,
gEH,
By Lemma 6, C(h) EH so that
a contradiction.
Lemma 9. There exists gEGn\H and bE(HC\Gn)\Z
such that
bl+«E(Hl+'T\Gn)\Z and bxE(H*r\Gn)\Z, where x=(l+g)~1.
Proof. By Lemma 4 there exists hiE(HC\Gn)\Z.
By Lemma 5
there exists gEGn\H such that (g, hi)E(Hr\Gn)\Z.
Let gi = (/h) 1+^
gi=ig, gi-i), hi=ig, hi-i) for i = 2, • • • , ra. By induction hiEHC\Gn
for i= 1, • ■ ■, n.
Now gi= ihi)1+"EH1+<ir\Gi because ÄiEG„CGi. Suppose for some
j-Kn
that g,-_i= ihj-iY+° E Hl+° C\ Gy_i. Then gj = ig, gj_i)
= (g- ihj-i)1+«)= ig, hj-iy+° = h)+'EH1+°. Now gEGnEGj and
ihj-iY+'EGj-i, so that ig, Ml'+'G^,
ThereforegjEHl+<T\G¡.By
induction gi = h\+'EHl+«r\d
for i=l,
• • • , ra.
Case I. ä„ is not central. Then gnEiH1+ar\Gn)\Z.
ment above with 1 + g replaced
by x will
The same argushow that
Qin)x
EiH1+<T\Gn)\Z. Letting b = hn, the lemma follows.
Case II. hn is central.
that hk~iEiHC\Gn)\Z
Then
there
exists an integer
k—l<n
such
and hk is central. If hk=ig, hk-i) = l, then
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i964]
A GENERALIZATIONOF THE CARTAN-BRAUER-HUATHEOREM
gEC(hk-i)EH,
contrary to Lemma 6. So hk9eI and commutes
hk-x and g. Therefore gk^l and gk commutes with gk-x and g.
215
with
Let tx= (g, gk-i + l), bx= (g, hk-x+ l), ti=(g, U-i), bt = (g, ô,_i) for
i=2, - - - , n + l. By induction, biEHi\Gi for i= 1, • • • , n. Then by
induction, t,= (bi)l+°EHl+°r\Gn
for i=l, ■ ■ ■ , n. If tmEZ lor some
m<n + l, then rm+i=l, and by Lemma 0, gk-x = (hk-x)l+a is algebraic
over Z; then hk-x is algebraic over Z, contrary to Lemma 8. So ¿,- is
not in Z for i = l, ■ ■ ■, n. Then b],+' = tnE(H1+<T\Gn)\Z.
commutes with hk-x and g, then
The argument of this paragraph
l+g replaced by x will prove that (bn)xE(HxC\Gn)\Z.
we are done.
Theorem
integer k.
Since hk
(hk)x commutes with (hk-i)x and g.
with gk-x replaced by (hk-i)x and
Letting b = b„,
1. A division ring has property Pk for any non-negative
Proof. Assume inductively that
Pn_i. Suppose A is a division ring
exist H, Gx, G2, • • ■ , G„ such that
sion ring of K which is invariant
where G„ is a noncentral subgroup
every division ring has property
without property P„. Then there
H is a noncentral proper subdiviunder G„<] • • • <3Gi<lGo = A',
of K'.
By Lemma 9 there exists gEG„\H, bEHC\Gn such that b1+s
EH1+°i\Gn and bxEHxC\Gn, where x=(l+g)~\
Let bx= c. Then
d = (b, bl+") = (c, b)l+»eiP+:
Since b1+"EGH, bEHi\Gn,
then
dEHC\Gn. Therefore dEZ by the corollary to Lemma 7.
If d=l, then (c, b) = l and cEC(b)EH by Lemma 6. Hence
b = cl+° EHr\GnC\
H1+" E Z, contradiction.
So ¿^1.
Let
ax=(b,b1+<' + l),dx=(c,b + l),ai=(b,ai-i),di=(c,di-x)lori
= 2, ■ ■ -,
n+l.
i=l,
Also let G„ = Gn+i. By induction ai=(dt)1+<'EHl+°r\Gi
for
■ ■ ■ , n + l. Now an+x=(b, an)EH, so that an+xEH1+oC\Gnr\H
EZ. Therefore (b, an+i) = 1. Since dEZ, then d commutes with b and
b1+". By Lemma 0, bl+s is algebraic over Z, and thus b is algebraic over
Z, contrary to Lemma 8.
Theorem 2. Let G be a noncentral subnormal subgroup of K'.
(i) For each noncentral kEK there exists gEG such that (g, k) is not
central;
(ii) C(G)=Z(K).
Proof.
Lemma 3 and Theorem 1 give (i). Then (ii) follows from (i).
Theorem 3. Let xEK\Z(K),
G a noncentral subnormal subgroup of
K', and M the conjugate class of x in G. Then M* = K.
Proof. By Theorem 1.
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216
C. J. STUTH
Theorem
groups.
[April
4. K' has no proper noncentral
subnormal
solvable sub-
Proof. Suppose G is a proper noncentral subnormal solvable subgroup of K'. There exist groups Gi, • • • , G» such that {1} = G„<] G„_i
<] • • • <¡d<\Go = G, where G< is the commutator
subgroup of G,_i
for i = 1, • • ■ , ra. Now G,- is a subnormal subgroup of K' for
t = 0, • • • , ra. By induction using Theorem 2, G,- is not central for
i=l, • • • , ra. But G„= {1} is central, a contradiction.
Theorem
5. Let A and B be noncentral subnormal subgroups of K'.
Then so is Ai\B,
Proof. Without loss of generality assume A and B are both rasubnormal in K'. Let^=^42n=
• ■ ■ =An<¡ ■ ■ • <\Ai<\A0 = K' and
B = B2n= • ■ ■ =Bn<¡ ■ ■ ■ <\Bx<\Bo = K'. Assume inductively that
for some i<2n that AjC\Bk is noncentral ¿-subnormal subgroup of
K', for all non-negative integers j and k such that j + k = i. Consider
A„r\Bh for positive integers g and h with g + h = i+l. Sinceg + Qi —l)
— ig~l)+h
= i, then there exists xEAgi\Bh-i
which is not central.
By Theorem 2 there exists an element yEAg-iC\Bh such that (x, y)
is not central. Since Agi^Bh is normal in both Ag(~\Bh-i and Ag_ir\Bh,
then (x, y) EAgi\Bh. So AgC\Bh is a noncentral (i+ l)-subnormal
subgroup of K'. By assumption Ai+iC\Bo = Ai+i and Ao(~\Bi+x = B¿+i are
noncentral
ii + l)-subnormal
subgroups of K'. By induction AC\B
= An(~\Bn in a noncentral
2ra-subnormal
subgroup
of K'.
Theorem 2 (ii) generalizes Huzurbazar
[4], and Theorem 3 generalizes Hua [2], Theorems 4 and 5 extend Scott's results in [6], The
following theorem extends the results about noncentral subnormal
subgroups of K' to subgroups of type I.
Theorem
6. Let H and L be subgroups of K', both of type I. Then
ii) CiH)=ZiK);
(ii) for each noncentral kEK
central ;
there is an hEH
such that ik, h) is not
(iii) H is not solvable;
(iv) H* = K;
(v) HCMis of type I.
Proof.
There exists noncentral
groups Gi, • • • , G„ such that H is
invariant under G = Gn<\ • • ■ <¡d<¡K'.
By Lemma 4, HC\G is noncentral and also HÍ^G<¡G. Then (i)-(iii) follow from Theorems 2 and
4 via HC\G. H* is invariant under G, hence H* = K by Theorem 1.
Let M be a noncentral subnormal subgroup of K' such that L is
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1964]
A GENERALIZATIONOF THE CARTAN-BRAUER-HUA
THEOREM
217
invariant under M. By Theorem 5, GC\M is a noncentral subnormal
subgroup of K'. H(~\L is invariant under GP\M. By Lemma 4, LC\M
is noncentral so that by Theorem 5, (HC\G)C\(LC\M)
is not central,
hence HC\L is not central.
Theorem
7. Let H be a subgroup of type I in K'. Then H is not
finite.
Proof. Suppose H is finite of minimum order. There exists a noncentral group G such that H is invariant under G. By Lemma 4,
HC\G is noncentral subnormal in K'. HEG since H is minimal. Hence
H is a noncentral «-subnormal subgroup of K', where « is minimal,
sayH=Hn<\H»-x<
■ ■ ■ <\Hx<lH0 = K'.
Case I. w=l. Now there exists xEH\Z(K).
Since H<]K', then x
has a finite number of conjugates in K', contrary to Theorem 4 of [6].
Case II. «> 1. Then by the minimality of n, there is yEHn-2 such
that H"^H.
H"<¡Hn-x and H« is not central, so by Theorem
5
Hi^H* is noncentral. Also HC\Hv<\Hn-x- This contradicts
the mini-
mality of H.
The author wishes to express his appreciation
to the National
Science Foundation for its support and to Professor W. R. Scott for
his suggestions, especially on Theorem 7.
Bibliography
1. I. N. Herstein and W. R. Scott, Subnormal subgroups of division rings, Canad.
J. Math. 15 (1963), 80-83.
2. L. K. Hua, Some properties of a sfield, Proc. Nat. Acad. Sei. U.S.A. 35 (1949),
533-537.
3. Nathan Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Publ. Vol. 37,
Amer. Math. Soc, Providence, R. I., 1956.
4. M. S. Huzurbazar,
The multiplicative group of a division ring, Dokl. Akad. Nauk
SSSR 131 (1960), 1268-1271.
5. Eugene Schenkman and W. R. Scott, A generalization of the Cartan-Brauer-Hua
theorem, Proc. Amer. Math. Soc. 11 (1960), 396-398.
6. W. R. Scott, On the multiplicative group of a division ring, Proc. Amer. Math.
Soc. 8(1957), 303-305.
University of Kansas and
East Texas State College
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