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On subnormal subgroups in general skew linear groups.

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УДК 512.74
Вестник СПбГУ. Сер. 1. 2013. Вып. 1
ON SUBNORMAL SUBGROUPS IN
GENERAL SKEW LINEAR GROUPS∗
Bui Xuan Hai1 , Nguyen Van Thin2
1. University of Science, VNU-HCMC, Vietnam,
Ph.D., bxhai@hcmus.edu.vn
2. University of Science, Vietnam,
Ph.D., nguyenvanthin20@gmail.com
This paper is dedicated to Professor Nikolai Vavilov on
the occasion of his sixtieth birthday.
1. Introduction. Let D be a division ring with center F and the multiplicative group
D∗ . In general, the structure of skew linear groups of degree n ≥ 1 over D is much less
known. The most important results concerning such groups can be found in [17] and [18].
In this paper, we study properties of subnormal subgroups of GLn (D) for n ≥ 1. Let N
be such a subgroup. We ask, under which conditions N must be in the center of GLn (D)?
Let us recall some history. In 1905 Wedderburn proved the famous result (now known as
Wedderburn’s “Little” Theorem): “Every finite division ring is commutative”. This means
that finite groups can not occur as multiplicative groups of non-commutative division
rings. This result stimulates many further investigations to generalize it. For instance,
the well-known theorem of Kaplansky [13, (15.15), p. 259] states that if D is radical over
F , then D = F ; L. K. Hua proved that if D∗ is solvable, then D is a field,... There is
a series of results of such a kind, known as commutativity theorems, even not only for
division rings, but for associative rings in general. We refer to [14] for more details. Now,
assume that N is a subnormal subgroup of D∗ = GL1 (D). C. J. Stuth [20] proved that if
N is solvable, then N ⊆ F . Evidently, Stuth’s result is a broad generalization of Hua’s
theorem. Recently, there are several results of such a kind appeared in the literature, see,
for example [1, 7–10, 16]. Hence, it is natural to consider the same problem for subnormal
subgroups of GLn (D) for n ≥ 2 and there are different results concerning this problem
obtained by several authors for last time (see, for example, [1–4, 15, 16], ...). In the present
paper, we give in addition, other conditions for N to be central.
Throughout this paper, D is a division ring with center F . We denote by D∗ the
multiplicative group of D and by D := [D∗ , D∗ ] the derived subgroup of D∗ . We say that
D is centrally finite if D is a finite dimensional vector space over F . If S is a non-empty
subset of D, then we denote by F [S] and F (S) the subring and the division subring of D
generated by S ∪ F respectively. If for any finite subset S of D, F (S) is a finite dimensional
vector space over F , then we say that D is locally finite. An element a ∈ D is algebraic
over F if there exists a non-zero polynomial f (t) ∈ F [t] such that f (a) = 0. A subset S
of D is algebraic over F if every a ∈ S is algebraic over F . An element x ∈ D is radical
over F , if there exists some positive integer n(x) depending on x such that xn(x) ∈ F .
∗ This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant 101.01-2011.16 and by Vietnam National University HoChiMinh City
(VNU-HCMC) under grant B2012-18-31.
c Bui Xuan Hai, Nguyen Van Thin, 2013
61
A non-empty subset S of D is radical over F if every element of S is radical over F .
If G is any group, then Z(G) is the center of G. The similar symbol is used for a ring.
In fact, if R is a ring, then Z(R) denotes the center of R. Recall that for n ≥ 2 we
have [GLn (D), GLn (D)] = SLn (D) unless the case n = 2 and D = 2 (the field of two
elements). Note that the center of Mn (D) is the set F I = {xI : x ∈ F }, where I is the
identity matrix and in the present paper we identify F I with F .
2. Results
Lemma 2.1. Let D be an infinite centrally finite division ring with center F . If N is a
finitely generated normal subgroup of GLn (D), n ≥ 2, then N is contained in F .
Proof. If N is not contained in F , then by [15, Theorem 11], SLn (D) ⊆ N . Since D
is infinite, SLn (D) is infinite and so N is infinite too. Now, by [2, Theorem 5], N is not
finitely generated, a contradiction.
Note that in [10], the notion of weakly locally finite division ring was firstly introduced
(see also Definition 1.1 in [4]). In fact, a division ring D is called weakly locally finite if for
every finite subset S of D the division subring of D generated by S is centrally finite. It was
proved in [10] that every locally finite division ring is weakly locally finite and the converse
is not true. Moreover, in [10], there is the example of a weakly locally finite division ring
that is not even algebraic over its center. Of course, every locally finite division ring is
algebraic (for instance, we note the longstanding conjecture [12], proposed by Kurosh in
1941, now known as Kurosh’s Problem for division rings which states that every algebraic
division ring is locally finite). Thus, the class of weakly locally finite division rings strictly
contains the class of locally finite division rings. In the following we mention some results
about these rings.
Theorem 2.2. Let D be an infinite weakly locally finite division ring with center F . If N
is a finitely generated subnormal subgroup of GLn (D), n ≥ 2, then N is contained in F .
Proof. If N is non-central, then by [15, Theorem 11], SLn (D) ⊆ N . Suppose that N is
generated by matrices A1 , . . . , Ak in GLn (D) and S is the set of all entries of all Aj , 1 ≤
j ≤ k. Since D is weakly locally finite, the division subring K of D generated by S
is centrally finite. By Lemma 2.1, N ⊆ Z(GLn (K)), so N is abelian and consequently,
SLn (D) is abelian too; a contradiction.
Remark 1. Theorem 2.2 is not longer true for finite division rings. In fact, if D is finite
then D is a field and for n ≥ 2 we have [GLn (D), GLn (D)] = SLn (D) (unless the case
n = 2 and D = 2 ) which is not contained in the center F = D. For the exceptional case,
we note that G = GL(2, 2) ∼
= S3 and SL(2, 2) ∼
= A3 , so the conclusion is also obvious.
In [7, Theorem 2.4], it was proved that if D is a division ring, algebraic over its center
F , then every locally solvable subnormal subgroup of D∗ is central. In the following we
show that this assertion remains also true for weakly locally finite division rings.
Theorem 2.3. Let D be a weakly locally finite division ring and N be a subnormal
subgroup of D∗ . If N is locally solvable, then N is contained in the center F of D.
Proof. Take any two elements x and y from N and put D1 = F (x, y), F1 = Z(D1 ) and
N1 = N ∩D1 . Since N1 is a subnormal locally solvable subgroup of D1∗ , by [7, Theorem 2.4],
62
N1 is contained in F1 . In particular, x and y commute with each other, so N is abelian.
Now, by [19, 14.4.4, p. 440], N is contained in F .
Recall Stuth’s theorem [20] which states that if N is a solvable subnormal subgroup
of D∗ , then N is central. So, in view of [7, Theorem 2.4] and Theorem 2.3 above, we have
a reason to pose the following conjecture.
Conjecture 1. Let D be an arbitrary division ring. If N is a locally solvable subnormal
subgroup of D∗ , then N is central.
As we see in the following theorem, the same assertion holds in GLn (D), n ≥ 2 with
only two exceptional “small” cases.
Theorem 2.4. Let D be a division ring and N a locally solvable subnormal subgroup of
GLn (D), n ≥ 2. Then, N is contained in the center F , unless the cases n = 2 and D = 2
or D = 3 .
Proof. Suppose that N is not contained in F . Then, by [15, Theorem 11], SLn (D) ⊆ N ,
so SLn (D) is locally solvable.
Case 1. n ≥ 3:
For 1 ≤ i = j ≤ n and x ∈ D, denote by Ei,j (x) the matrix with x on the position
(i, j), ones on the main diagonal and 0 elsewhere. Consider the set
S = {E12 (1), E21 (1), E31 (1), E13 (1), E23 (1), E32 (1)}
and suppose that H = S is the subgroup of SLn (D) generated by S. Then, H is solvable.
On the other hand, since [Eij (1), Ejk (1)] = Eik (1) for any distinct i, j, k, we have S ⊆ H ,
hence H = H . The last equality shows that H is unsolvable, a contradiction.
Case 2. n = 2:
Subcase 1. CharD = p : Since PSL(2, p) is a non-abelian simple group, it is unsolvable. Hence, SL(2, p) is unsolvable finite subgroup of SLn (D) that is a contradiction.
Subcase 2. CharD = 0 :
Firstly, we note that the subgroup
H = E12 (1/2), E21 (1/2), E12 (3), E21 (1/3)
is solvable. By direct calculation, we have:
2
0
a=
= E21 (−1/2)E12 (1)E21 (1)E12 (−1/2) ∈ H,
0 1/2
3
0
b=
= E21 (−2/3)E12 (1)E21 (2)E12 (−1/3) ∈ H,
0 1/3
E12 (1) = [a−1 , E12 (−1)]3 [E12 (−1), b−1 ] ∈ H ,
E21 (1) = [b, E21 (1)][E21 (1), a]3 ∈ H ,
E12 (1/2) = [a−1 , E12 (−1/2)]E12 (−1) ∈ H ,
E12 (1/3) = [E12 (−1/3), b−1 ]E12 (3) ∈ H ,
E21 (1/2) = E21 (−1)[E21 (1/2), a] ∈ H ,
E21 (1/3) = E21 (3)[b, E21 (1/3)] ∈ H .
Thus, H = H and this shows that H is unsolvable, a contradiction.
63
Remark 2. The cases n = 2 and D = 2 or D = 3 are really exceptional. In fact, the
subgroups SL(2, 2) and SL(2, 3) are non-central solvable subnormal subgroups in GL(2, 2)
and GL(2, 3) respectively.
Corollary 2.5. Let D be a division ring and N a locally nilpotent subnormal subgroup of
GLn (D), n ≥ 1. Then, N is contained in the center F , unless the cases n = 2 and D = 2
or D = 3 .
Proof. In fact, the case n = 1 is Theorem 2.2 in [7]. For n ≥ 2, the conclusion follows
from Theorem 2.4.
Theorem 2.6. let D be a weakly locally finite division ring and suppose that N is a
subnormal Engel subgroup of D∗ . Then, N is contained in the center F of D.
Proof. Consider an arbitrary finite subset S of N and the subgroup H = S of N
generated by S. If any two elements of S commute with each other, then H is abelian.
Otherwise, put D1 = F (S) and denote by F1 the center of D1 . Then, it is clear that
2 ≤ m = [D1 : F1 ] < ∞. So, H can be viewed as a subgroup of GLm (F1 ). Since N is an
Engel subgroup, H is an Engel subgroup too. By the result of Gruenberg [5], H coincides
with its Hirsch-Plotkin radical, so H is nilpotent. Consequently, N is locally nilpotent.
Now, by Corollary 2.5, N ⊆ F .
Theorem 2.7. Assume that a division ring D with center F is different from
and n ≥ 2. If N is a subnormal Engel subgroup of GLn (D), then N ⊆ F.
2
and
3,
Proof. Suppose that N ⊆
F . Then, by [15, Theorem 11], SLn (D) ⊆ N . Put
⎞
⎞
⎛
⎛
1 −1
2−1
⎠ and y = ⎝
⎠.
2
x=⎝
1
In−2
In−2
For any integer k ≥ 1 we have
⎛
[x, k y] = ⎝
⎞
1 −3k
1
In−2
⎠ = In .
This shows that SLn (D) is not Engel subgroup, a contradiction. Hence, N ⊆ F .
Theorem 2.8. Let D be a division ring with center F and suppose that N is a subnormal
subgroup of GLn (D), n ≥ 2. Assume that D is not a field of characteristic p, algebraic
over p . If N is radical over F , then N ⊆ F .
Proof. Suppose that N ⊆ F . Then, by [15, Theorem 11], SLn (D) ⊆ N .
Case 1. D is non-commutative
For any 1 = d ∈ D we have A = diag[d, 1, . . . , 1] ∈ SLn (D), so there exists some integer k such that Ak = diag[dk , 1, . . . , 1] = In . Hence, d is torsion and by [[14], Theorem 8],
D ⊆ F . So, D∗ is solvable, and by [19, 14.4.4, p. 440], D is commutative, a contradiction.
Case 2. D is commutative
Suppose that CharD = 0. Then, we have A = diag[2, 1/2, 1, . . . , 1] ∈ SLn (D) ⊆ N , so
A is radical over F . But this is a contradiction, since Ak ∈ F for any integer k ≥ 1. Hence,
it remains to consider the case CharD = p > 0. Suppose that D is not algebraic over p .
64
Then, there exists some element a ∈ D such that ak = 1 for all k ≥ 1. Hence, the matrix
diag[a, a−1 , 1, . . . , 1] is not radical over F although it is in SLn (D), a contradiction.
Thus, in any case we have N ⊆ F and the proof is now complete.
We note that in the theorem above the condition that D is not a field of characteristic
p, algebraic over p is really necessary (see Remark 3 below).
Corollary 2.9. If n ≥ 2 and SLn (D) contains some proper subgroup of finite index, then
D must be a field of characteristic p, algebraic over p .
Proof. Assume that N is a proper subgroup of finite index in SLn (D). Then, H =
−1
N g is the normal subgroup of finite index in SLn (D), say, [SLn (D) : H] =
g∈SLn (D) g
r < ∞. By [6, 2.2.10, p. 79] and [6, 2.2. 13, p. 80], H ⊆ F . Since g r ∈ H, ∀g ∈ SLn (D), it
follows that SLn (D) is radical over F . Now, the conclusion follows from Theorem 2.8.
Corollary 2.10. If n ≥ 2 and SLn (D) contains some non-central F C-element, then D
must be a field of characteristic p, algebraic over p .
Proof. Assume that x ∈ SLn (D) is a non-central F C-element. Then, [SLn (D) :
CSLn (D) (x)] = r < ∞ and r ≥ 2. Now, the conclusion follows from Corollary 2.9.
Corollary 2.11. Let D be a division ring with center F and suppose that N is a subnormal
subgroup of GLn (D), n ≥ 1. Assume that D is not a field of characteristic p, algebraic
over p . If N is locally finite, then N ⊆ F . Moreover, if n = 1, then the conclusion holds
without any constrain for D.
Proof. If n ≥ 2, then the conclusion follows from Theorem 2.8. The case n = 1 follows
from [8, Theorem 2.3].
Theorem 2.12. Let D be a division ring with center F and suppose that N is a subnormal
subgroup of GLn (D), n ≥ 1. Assume that D is not a field of characteristic p, algebraic
over p . If N is an F C-group, then N ⊆ F . Moreover, if n = 1, then the conclusion holds
without any constrain for D.
Proof. If n = 1, then N ⊆ F by [8, Theorem 3.1].
Now, for n ≥ 2, suppose that N ⊆ F . Then, by [15, Theorem 11], SLn (D) ⊆ N , so
SLn (D) is an F C-subgroup.
Case 1. D is non-commutative
d
Take an element d = 1 in D and consider a matrix A =
in SLn (D).
In−1
Since SLn (D) is an F C-subgroup, the following subset is finite
a
−1 In−1
d
In−1
a
In−1
−1
a da
|a ∈ D =
In−1
a ∈ D .
Hence, D is an F C-subgroup and by [8, Theorem 3.1], D ⊆ F , so D∗ is solvable,
and consequently D is commutative, a contradiction.
65
Case 2. D is commutative
Assume that CharD = 0. Then, the following set
⎛
1
⎝ k
⎞−1 ⎛
1
⎠
In−2
⎛
⎝
⎞⎛
2
1/2
2
= ⎝ −3k/2 1/2
In−2
In−2
1
⎠⎝ k
⎞
⎞
⎠=
1
In−2
⎠ k ∈ is infinite that is a contradiction since SLn (D) is an F C-subgroup.
Hence, it remains to consider the case CharD = p. By supposition, D is not algebraic
over p , so there exist a, b ∈ D∗ such that a is not algebraic over p and b = b−1 . Then,
the following set
⎞−1 ⎛
⎞⎛
⎞
⎛
b
1
1
⎠ ⎝
⎠ ⎝ ai 1
⎠ i ∈ ⎝ ai 1
b−1
In−2
In−2
⎞ In−2
⎛
b
⎠ i ∈ is infinite, that is a contradiction.
= ⎝ ai (b−1 − b) b−1
In−2
Thus, in any case we have a contradiction; hence N ⊆ F , so the proof is now complete.
Remark 3. 1) The condition that D is not a field of characteristic p, algebraic over p
in Theorem 2.8 and Theorem 2.12 is really necessary. In fact, suppose that F is a field of
characteristic p, algebraic over p and n ≥ 2. Consider an arbitrary matrix A = (aij ) ∈
SLn (F ) and assume that K is the subfield of F generated by all aij over p . Then K is a
finite field and it follows that SLn (K) is a finite group. Hence, A is a torsion element and
consequently, SLn (F ) is a torsion non-central normal subgroup of GLn (F ).
2) If F is a finite field, then the conclusion in Corollary 2.11 is no longer true. In
fact, this can be seen from the fact that for n ≥ 2, SLn (F ) is a finite non-central normal
subgroup of GLn (F ).
Corollary 2.13. Let D be a division ring with center F and suppose that N is a subnormal
subgroup of D∗ . If a ∈ N is an F C-element in N , then a ∈ F .
Proof. Clearly we can suppose that D is non-commutative. The set S = {ag |g ∈ N } is
finite, so we can write S = {ag1 , . . . , agt } for some positive integer t. Clearly, the subgroup
H = S is normal in N . For x ∈ H, we have the following expression
x = aε1 gj1 . . . aεk gjk , with εi = ±1 and gji ∈ S.
Choose k as small as possible in the above expression of x. Then, the number of
elements xh , h ∈ H does not exceed tk , so H is an F C-group. Now, by [8, Theorem 3.1],
H ⊆ F and so a ∈ F .
66
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Статья поступила в редакцию 26 июня 2012 г.
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