Finitely Generated Subnormal Subgroups of GL_{n}(D) Are Central ^{∗}

M. Mahdavi-Hezavehi M. G. Mahmudi, S. Yasamin ^{†}

Abstract Let D be an infinite division algebra of finite dimension over its centre. Assume that N is a subnormal subgroup of GL_{n}(D) with n ≥ 1. It is shown that if N is finitely generated, then N is central.

Let D be an infinite division algebra of degree m over its centre Z(D) = F. Denote by D′ the commutator subgroup of the multiplicative group D^{∗} = D − {0}. The aim of this note is to investigate the structure of finitely generated subnormal subgroups of GL_{n}(D) with n ≥ 1. Assume that n ≥ 2 and N is a normal subgroup of GL_{n}(D). It is shown in [1] that if N is finitely generated, then N is central. A similar result for finitely generated normal subgroups is obtained for the case n = 1 in [2]. Here we shall generalize some of the main results appeared in [1] and [2] to subnormal subgroups of GL_{n}(D) with n ≥ 1. To be more precise, assume that N is a subnormal subgroup of GL_{n}(D) with n ≥ 1. It is proved that if N is finitely generated, then N is central. Using this, it is also shown that GL_{n}(D)/Z(GL_{n}(D)) contains no non-trivial finitely generated subnormal subgroups. Furthermore, given an infinite subnormal subgroup N of GL_{n}(D), it is proved that N contains no finitely generated maximal subgroups. Therefore, GL_{n}(D) contains no finitely generated maximal subgroups. The reader may consult [5], [6], [7], and the references thereof for more recent results on multiplicative subgroups of GL_{n}(D). For convenience we shall deal with the case n = 1 separately. Our key result is the following

Theorem 1. Let D be a finite dimensional division algebra with centre F. Then any finitely generated subnormal subgroup of D^{∗} is central.

Proof. We first claim that if D^{∗} contains a non-central finitely generated subgroup N, which is subnormal, then F is finitely generated over its prime subfield.

∗Key words: Division ring, subnormal, finitely generated.

†AMS (1991) Subject classification : 15A33, 16K40

To see this, assume that E is the division algebra generated by all elements u of N. Since E is invariant under all inner automorphisms x → uxu^{−1}, it follows from a result of Stuth (cf. [8, p. 439]) that either E is central or E = D. If E is central, then N is central, a contradiction. Thus, we may assume that E = D. Suppose that [D : F] = n and consider the regular matrix representation of D^{∗} in GL_{n}(F). Since N is finitely generated, there exist matrices A_{1}, …, A_{k} ∈ GL_{n}(F) such that N =< A_{1}, …, A_{k} >. Let Λ be the set of all elements in F occurring as the entries of A_{i} and A_{i}^{−1}, i = 1, …, k. If H is the subring generated by N, then we have H ⊂ GL_{n}(P (Λ)), where P (Λ) is the subfield of F generated by Λ over the prime subfield P. Now, since E ⊂ GL_{n}(P (Λ)) we have aI ∈ GL_{n}(P (Λ)), for any a ∈ F^{∗} and so a ∈ P (Λ). Hence, F = P (Λ) and the claim is established. To proceed the proof, let N be a non-central finitely generated subgroup which is subnormal in D^{∗}. Since F is finitely generated over the prime subfield P, by Noether Normalization Lemma, there exist elements r_{1}, …, r_{s} ∈ F which are algebraically independent over P such that [F : P (r_{1}, …, r_{s})] < ∞, where s denotes the transcendency degree of F over P. For convenience set r_{s} = y, L = P (r_{1}, …, r_{s}), and K = P (r_{1}, …, r_{s−1}) with K = P if s = 0. Since [F : L] < ∞ we obtain k = [D : L] < ∞ and so D^{∗} has a matrix representation in GL_{k}(L). We may now consider two cases: Case (1): s = 0. If CharD = p > 0, then D is algebraic over a finite field and consequently , by a result of Jacobson (cf. [4]), we conclude that D is commutative which is in contradiction with the fact that N is non-central. Thus, we may assume that P = Q, the field of rational numbers. For each a, b ∈ N and x ∈ L, set c_{1} = c_{1}(a, b, x) = (b + x)a(b + x)^{−1}, and for m > 1 define c_{m} inductively by c_{m} = c_{m−1}bc_{m−1}^{−1}. Since N is subnormal in D^{∗} there exists a natural number r such that N = N_{r} ⊲ N_{r−1} ⊲ ⋯ ⊲ N_{1} ⊲ D^{∗}. Thus, c_{1} ∈ N_{1} and by induction we conclude that c_{r} ∈ N_{r}. We now claim that for each i we have c_{i} = (b + x)w_{i}(a, b)(b + x)^{−1}, where w_{i}(a, b) is a reduced word in a, a^{−1}, b, b^{−1} whose first and last alphabets are a or a^{−1}, respectively. In fact, for i = 1 we have c_{1} = (b + x)a(b + x)^{−1} and if c_{i} = (b + x)w_{i}(a, b)(b + x)^{−1}, then, by induction, we conclude that c_{i+1} = c_{i}bc_{i}^{−1} = [(b+x)w_{i}(a, b)(b+x)^{−1}]b[(b+x)w_{i}(a, b)^{−1}(b+x)^{−1}] = (b + x)[w_{i}(a, b)bw_{i}(a, b)^{−1}](b + x)^{−1}, and since the first and the last alphabets of w_{i}(a, b)bw_{i}(a, b)^{−1} are a or a^{−1}, the claim is established. Since N is subnormal in D^{∗} and [D : F] < ∞, by a result of Goncalves (cf. [3]), N contains a noncyclic free subgroup G, say. Take a, b to be the generators of G, and denote their matrix representations in GL_{k}(L) by A and B, respectively. Since N is finitely generated, by the argument used above, we conclude that there is a set Λ = {m_{1}/n_{1}, …, m_{t}/n_{t}} ⊂ Q such that each element of N has a matrix representation in GL_{k}(Z[Λ]), where Z is the ring of integers. As observed above, since N is subnormal in D^{∗}, for each element x ∈ Q we have c_{r} = c_{r}(a, b, x) ∈ N. Thus, c_{r}(A, B, xI) = (B + xI)w_{r}(A, B)(BI + x)^{−1} ∈ N. Since det(B + xI) is a polynomial in x of degree k, and for each 1 ≤ i, j ≤ k, the (i, j)-th entry of (B + xI)^{−1} is of the form f_{ij}(x)/g(x) ∈ Q(x), where deg g(x) = k, deg f_{ij}(x) ≤ k − 1, we conclude that the (i, j)-th entry of the matrix c_{r}(A, B, xI) is of the form f_{ij}(x)/g(x), where for each 1 ≤ i, j ≤ k, we have deg f_{ij}(x) ≤ k. If for each 1 ≤ i, j ≤ k, there are rational numbers q_{ij} such that for any x ∈ Q, f_{ij}(x)/g(x) = q_{ij}, then c_{r}(a, b, x) is independent of x and so putting x = 0, and x = 1 we obtain c_{r}(a, b, 0) = c_{r}(a, b, 1). This implies that bw_{r}(a, b)b^{−1} = (b + 1)w_{r}(a, b)(b + 1)^{−1}, and consequently, bw_{r}(a, b) = w_{r}(a, b)b. Since the first and the last alphabets of w_{r}(a, b) are a or a^{−1}, respectively, this gives us a non-trivial relation between a and b which is a contradiction to the fact that G is free. Thus there exists an entry of c_{r}(A, B, xI), say (i, j)-th which depends on x. Put f_{ij}(x) = Σ_{i=0}^{k}a_{i}x^{i}, g(x) = x^{k} + Σ_{i=0}^{k−1}b_{i}x^{i}. Thus for each x ∈ Q we have f_{ij}(x)/g(x) ∈ Z[Λ]. If a_{k} = m_{t+1}/n_{t+1}, then for each x ∈ Q we obtain f_{ij}(x)/g(x)−a_{k} ∈ Z[Λ∪{m_{t+1}/n_{t+1}}]. So there exists a polynomial f(x) ∈ Q[x] such that deg f(x) ≤ k − 1 and for each x ∈ Q we have f(x)/g(x) ∈ Z[Λ ∪ {m_{t+1}/n_{t+1}}] . Multiplying f(x) and g(x) by suitable scalars, we may assume that f(x), g(x) ∈ Z[x]. Put f(x) = Σ_{i=0}^{k−1}a′_{i}x^{i} ∈ Z[x], g(x) = Σ_{i=0}^{k}b′_{i}x^{i}. Since det B ≠ 0, we may assume that b′_{0} ≠ 0. Now, change the variable x to b′_{0}x to obtain f_{1}(x), g_{1}(x) ∈ Z[x], such that deg g_{1} = k, degf_{1} ≤ k − 1, where the constant term of g_{1}(x) is 1, and for each x ∈ Q we have, f_{1}(x)/g_{1}(x) ∈ Z[Λ ∪ {m_{t+1}/n_{t+1}}]. Assume that S = {p_{1}, …, p_{l}} be the set of all primes occurring in the factorizations of n_{1}, …, n_{t+1} into prime numbers. For each natural number r, put x_{r} = (p_{1}p_{2} … p_{l})^{r}. Since deg f_{1} < deg g_{1}, for a large enough number r, we obtain that f_{1}(x_{r})/g_{1}(x_{r}) < 1. On the other hand, for each r ≥ 1, and each 1 ≤ i ≤ l, g_{1}(x_{r}) and p_{i} are coprime, that is, (g_{1}(x_{r}), p_{i}) = 1. It is not hard to see that if u/v ∈ Z[m_{1}/n_{1}, …, m_{t+1}/n_{t+1}] with (u, v) = 1, then each prime factor of v belongs to S. Now since f_{1}(x_{r})/g_{1}(x_{r}) ∈ Z[m_{1}/n_{1}, …, m_{t+1}/n_{t+1}] and for each 1 ≤ i ≤ l, r ≥ 1, (g_{1}(x_{r}), p_{i}) = 1, we reach a contradiction, and so the result follows in this case.

Case (2): s > 0. Since N is finitely generated, by the argument used in the first case, we conclude that there is a set Λ = {m_{1}/n_{1}, …, m_{t}/n_{t}} ⊂ L, where m_{i}, n_{i} ∈ K[y] such that each element of N has a matrix representation in GL_{k}(K[y][Λ]), and K[y][Λ] is the subring of L generated by Λ over K[y]. On the other hand, N is subnormal in D^{∗} and thus by the same argument used in the first case, there exists an entry of c_{r}(A, B, xI), say (i, j)-th which depends on x. Put f_{ij}(x) = Σ_{i=0}^{k}a_{i}x^{i}, g(x) = x^{k} + Σ_{i=0}^{k−1}b_{i}x^{i}. Thus for each x ∈ L we have f_{ij}(x)/g(x) ∈ K[y][Λ]. If a_{k} = m_{t+1}/n_{t+1}, then for each x ∈ L we obtain f_{ij}(x)/g(x) − a_{k} ∈ K[y][Λ ∪ {m_{t+1}/n_{t+1}}]. So there exists a non-zero polynomial f(x) = f_{ij}(x) − a_{k}g(x) ∈ L[x] such that deg f(x) ≤ k − 1 and for each x ∈ L we have f(x)/g(x) ∈ K[y][Λ ∪ {m_{t+1}/n_{t+1}}] . Multiplying f(x) and g(x) by suitable elements of K[y], we may assume that f(x), g(x) ∈ K[y][x]. Put f(x) = Σ_{i=0}^{k−1}a′_{i}x^{i}, g(x) = Σ_{i=0}^{k}b′_{i}x^{i}. Since det B ≠ 0, we may assume that b′_{0} ≠ 0. Now, change the variable x to b′_{0}x to obtain f_{1}(x), g_{1}(x) ∈ K[y][x], such that deg g_{1} = k, degf_{1} ≤ k − 1, and the constant term of g_{1}(x) is 1. Further, for each x ∈ L we have f_{1}(x)/g_{1}(x) ∈ K[y][Λ ∪ {m_{t+1}/n_{t+1}}]. Assume that S = {p_{1}, …, p_{l}} is the set of all irreducible polynomials occurring in the factorizations of n_{1}, …, n_{t+1} into irreducible polynomials. For each natural number r, put x_{r} = (p_{1}p_{2} … p_{l})^{r}. Since deg f_{1} < deg g_{1}, for a large enough number r, the degree of the denominator of f_{1}(x_{r})/g_{1}(x_{r}) with respect to y is greater than that of the nominator. On the other hand, for each r ≥ 1, and each 1 ≤ i ≤ l, g_{1}(x_{r}) and p_{i} are coprime, that is, (g_{1}(x_{r}), p_{i}) = 1. It is not hard to see that if u/v ∈ K[y][m_{1}/n_{1}, …, m_{t+1}/n_{t+1}] with (u, v) = 1, then each irreducible factor of v belongs to S. Now since f_{1}(x_{r})/g_{1}(x_{r}) ∈ K[y][m_{1}/n_{1}, …, m_{t+1}/n_{t+1}] and for each 1 ≤ i ≤ l, r ≥ 1, (g_{1}(x_{r}), p_{i}) = 1, we arrive at a contradiction, and so the result follows.

As a consequence of the above theorem, we have the following

Corollary 1. Let D be an infinite division algebra of finite dimension over its centre F. Assume that N is a subnormal subgroup of GL_{n}(D) with n ≥ 1. If N is finitely generated, then N ⊂ F^{∗}.

Proof. The case n = 1 follows from Theorem 1. Now, consider the case n > 1. By Theorem 11 of [5], we have SL_{n}(D) ⊂ N. Thus N is normal in GL_{n}(D). Finally, using Theorem 5 of [1], we obtain the result for n ≥ 2.

Corollary 2. Let D be an infinite division algebra of finite dimension over its centre F and n ≥ 1. Then GL_{n}(D)/Z(GL_{n}(D)) contains no non-trivial finitely generated subnormal subgroups.

Proof. Identify Z(GL_{n}(D)) with F^{∗} and let N/F^{∗} be a finitely generated subnormal subgroup of GL_{n}(D)/F^{∗}. Let x_{1}F^{∗}, …, x_{r}F^{∗} be a set of generators of N/F^{∗}. If we set G =< x_{1}, …, x_{r} >, the group generated by x_{1}, …, x_{r}, we conclude that N = GF^{∗}. Thus, N′ = G′ and consequently G is normal in N. This implies that G is subnormal in GL_{n}(D). Now, using Corollary 1, we conclude that N = F^{∗} which completes the proof.

Corollary 3. Let D be a division algebra of finite dimension over its centre F and n ≥ 1. Assume that N is an infinite subnormal subgroup of GL_{n}(D). Then N contains no finitely generated maximal subgroups. In particular, if D is infinite, then GL_{n}(D) contains no finitely generated maximal subgroups.

Proof. Assume that M is a maximal subgroup of N which is finitely generated. Then for any x ∈ N\M we have < M, x >= N. This implies that N is finitely generated which contradicts Corollary 1.

Bearing in mind the fact that Z(SL_{n}(D)) = SL_{n}(D) ∩ Z(GL_{n}(D)), we may conclude

Corollary 4. Let D be a division algebra of finite dimension over its centre F and n ≥ 1. Assume that N is an infinite subnormal subgroup of SL_{n}(D). If N is finitely generated, then N ⊂ Z(SL_{n}(D)). Furthermore, if D is infinite, then P SL_{n}(D) = SL_{n}(D)/Z(SL_{n}(D)) contains no non-trivial finitely generated subnormal subgroups.

Proof. The first part follows from Corollary 1 and the fact that Z(SL_{n}(D)) = SL_{n}(D) ∩ Z(GL_{n}(D)). The proof of the final part is similar to that of Corollary 2.

It is believed that the condition on D being of finite dimension over its centre is superfluous in all the above results.

The authors would like to thank the referee for his constructive comments. They would also thank the Research Council of Sharif University of Technology for support.

References

[1] Akbari, S., Mahdavi-Hezavehi, M., Normal Subgroups of GL_{n}(D) Are not Finitely Generated, Proc. Amer. Math. Soc., To appear.

[2] Akbari, S., Mahdavi-Hezavehi, M., Mahmudi, M. G., Maximal Subgroups of GL_{1}(D), Journal of Alg., To appear.

[3] Goncalves, J. Z., Free Groups in Subnormal Subgroups and the Residual Nilpotence of the Group of Units of Group Rings, Can. Math. Bull. 27(1984), 365-370.

[4] Jacobson, N., Structure of Rings, Coll., Pub., Vol. 37, Amer. Math. Soc., Providence, R.I., (1956).

[5] Mahdavi-Hezavehi, M., and Akbari, S., Some Special Subgroups of GL_{n}(D) , Algebra Colloquium 5:4 (1998), 361-370.

[6] Mahdavi-Hezavehi, M., On Derived Groups of Division Rings, Comm. Algebra, Vol.23(3), 913-926, (1995).

[7] Mahdavi-Hezavehi, M., Akbari, S., Mehraabaadi, M., and Hajie-Abolhassan, H., On Derived Groups of Division Rings II, Comm. Algebra, Vol. 23, No. 8, (1995), 2881-2887.

[8] Scott, W. R., Group Theory, Dover Publication, INC, 1987.

M. Mahdavi-Hezavehi, mahdavi@math.sharif.ac.ir
M. G. Mahmudi, gholamza@math.sharif.ac.ir
S. Yasamin, yasimin@math.sharif.ac.ir

Department of Mathematical Sciences Sharif University of Technology P.O.Box 11365-9415 Tehran, Iran