Mathematics

A. I. LIKHTMAN

ON SUBGROUPS OF THE MULTIPLICATIVE GROUP OF A DIVISION RING

(Presented by Academician A. I. Mal’cev, 10 III 1965)

Let \(D\) be an arbitrary division ring. We shall denote the multiplicative group of this division ring by \(D^*\); the center of the division ring \(D\) by \(Z\); the multiplicative group of the field \(Z\) by \(Z^*\); and the identity element of the group \(D^*\) by \(e\).

Herstein \((^5)\) proved some theorems on finite subgroups of the group \(D^*\). Amitsur \((^4)\) described all finite groups that can be embedded in the multiplicative group of some division ring. In particular, every finite \(p\)-subgroup of the group \(D^*\) is either a cyclic group or a group of generalized quaternions.

D. A. Suprunenko informed the author that for \(p \ne 2\) the Sylow \(p\)-subgroups of the multiplicative group of a division algebra finite-dimensional over its center are conjugate, while Sylow \(2\)-subgroups are conjugate in a number of special cases. In particular, in order to prove the conjugacy of Sylow \(2\)-subgroups of the multiplicative group of a finite-dimensional division algebra it is necessary to show that, if one of the Sylow \(2\)-subgroups is a group of generalized quaternions with defining relations \(a^{2^n} = 1\), \(b^2 = a^{2^{n-1}}\), \(b^{-1}ab = a^{-1}\), then any other Sylow \(2\)-subgroup can be neither a cyclic group of order greater than \(2^n\) nor a group of type \(2^\infty\).

We shall construct an example of a division ring \(D\) whose multiplicative group \(D^*\) contains, as Sylow \(2\)-subgroups, a cyclic group \(F_1\) of order 8 and a group \(F_2\) of generalized quaternions. The division ring \(D\) will have dimension 16 over its center \(Z\), which is the field of rational functions \(R(t)\) in one variable over the field of rational numbers \(R\).

Let \(R(x)\) be the field of rational functions over \(R\); let \(s\) be the automorphism of the field \(R(x)\) generated by the mapping \(x \to -x^{-1}\); let \(A = R(x)[Y,s]\) be the set of noncommutative polynomials over \(R(x)\) of the form \(a_0 + a_1Y + \ldots + a_mY^m\), where \(a_i \in R(x)\), \(Y\) is an indeterminate, and multiplication and addition are defined in the natural way by means of the relation
\[ (Y^k\alpha)(Y^l\beta)=Y^{k+l}(\alpha s^l)\beta. \]
It is known that \(A\) is an associative ring. It is clear that the element \(Y^2\) lies in the center of \(A\). It is not difficult also to see that the quotient ring \(Q\) of the ring \(A\) by the principal ideal \(B=(Y^4+1)A\) contains a subfield, isomorphic to \(R(x)\), which we shall also denote by \(R(x)\). We denote by \(a\) the residue class corresponding to the element \(Y\) under the natural homomorphism of \(A\) onto \(Q\). Denote by \(R_1\) the field \(R(t)\), where \(t=x^{-1}a^2+xa^{-2}\). The element \(t\) is transcendental over \(R\). It is not difficult to show that \((Q:R_1)=8\).

Lemma 1. Let \(t_1=t-2\), \(t_2=t+2\). Denote by \(\rho_1,\rho_2\) the \(p\)-adic norms of the field \(R_1\) determined by the elements \(t_1,t_2\), respectively. Let \(\overline{R}_1\) be the completion of the field \(R_1\) with respect to the norm \(\rho_1\); let \(\overline{\overline{R}}_1\) be the completion of the field \(R_1\) with respect to the norm \(\rho_2\), \(\overline{Q}=Q_{R_1}\otimes \overline{R}_1\), \(\overline{\overline{Q}}=Q_{R_1}\otimes \overline{\overline{R}}_1\). Then the algebras \(\overline{Q}\) and \(\overline{\overline{Q}}\) are division algebras of dimension 8 over the fields \(\overline{R}_1\) and \(\overline{\overline{R}}_1\), respectively.

Consider the subgroup \(G_1\) of the group \(Q^*\) generated by the elements \(a\) and \(x\). Each element \(g\) of the group \(G_1\) is uniquely represented in the form \(g=a^i x^j\), where \(0 \le i < 8\), and \(j\) is arbitrary. Let \(\varphi\) be a mapping of the group \(G_1\) into itself,

constructed as follows: \(\varphi(e)=e\), \(\varphi(x)=x^{-1}\), \(\varphi(a)=ax^{-1}\), \(\varphi(a^i x^j)=\varphi(a)^i\varphi(x)^j\). It can be shown that \(\varphi\) is an automorphism of the group \(G_1\). The automorphism \(\varphi\) extends to an automorphism of the algebra \(Q\) over \(R_1\), and then to an automorphism of \(\overline Q\) over \(\overline R_1\), which we shall also denote by \(\varphi\).

Consider the polynomial ring \(\overline Q[Z,\varphi]\), where \(\alpha Z=Z(\alpha\varphi)\) for \(\alpha\in \overline Q\). Denote by \(Q_1\) the quotient ring of the ring \(\overline Q[Z,\varphi]\) by the principal ideal \((Z^2+1)\) (it is easy to see that the polynomial \(Z^2+1\) lies in the center of \(\overline Q[Z,\varphi]\)).

Lemma 2. The ring \(Q_1\) is a division algebra of dimension 16 over the field \(\overline R_1\).

Denote by \(b\) the image of the element \(Z\) under the canonical homomorphism \(\overline Q[Z,\varphi]\) onto \(Q_1\). Let \(G\) be the group generated by \(G_1\) and \(b\); \(D\) the linear span of the group \(G\) over \(R\). It follows from Lemma 2 that \(D\) is a division algebra over \(R_1\) of dimension 16. It is not hard to show that the center of \(D\) coincides with the field \(R\).

The group \(D^*\) contains a cyclic subgroup \(F_1\) of order eight, generated by the element \(a\), and a quaternion subgroup \(F_2\), generated by the elements \(a^2\) and \(b\).

Lemma 3. The subgroups \(F_1\) and \(F_2\) of the group \(D^*\) are Sylow 2-subgroups.

In the proof of Lemma 3 one essentially uses the fact that the algebra \(\overline Q\) is a division algebra over \(\overline R_1\).

As has already been noted, in our example the center of the division ring \(D\) is the field of rational functions over \(R_1\).

Theorem 1. Let \(D\) be a finite-dimensional division algebra over a field \(T\) of algebraic numbers. Then the Sylow \(p\)-subgroups of the group \(D^*\), for a given \(p\), are conjugate.

From the results of (²) it follows that all solvable normal divisors of the group \(D^*\) lie in the center. Herstein (⁷) showed that all locally nilpotent normal divisors of the group \(D^*\) lie in the center. In (³) it is proved that in the factor group \(D^*/Z^*\) there are no nontrivial locally finite normal divisors. At the same time, in (¹) an example is given of a division ring whose multiplicative group contains an infinite descending chain of invariant subgroups**.

Proposition 1. Let \(D\) be a division algebra of dimension \(q^m\) over its center \(T\), which is algebraic over the field \(R\) of rational numbers. Then, if \(q^m\ne 4\), the group \(D^*\) contains a noncentral normal divisor \(F\) for which the intersection of all terms of the lower central series is equal to the identity subgroup.

Proposition 1 is used to prove Theorem 2, from which, in particular, there follows a series of known facts about normal divisors of the multiplicative group of a division ring and some new results.

Theorem 2. Let \(D\) be an arbitrary division ring, and let \(G\) be a normal divisor of the group \(D^*\). If for any elements \(x,y\) of \(G\) there exists a number \(n\) (depending on \(x,y\)) such that all right-normed \(n\)-fold commutators of the subgroup \(G_1\) generated by \(x\) and \(y\) have finite orders, then \(G\subset Z\).

Received
3 III 1965

References

¹ N. Jacobson, Structure of Rings, Moscow, 1961.
² W. R. Scott, Proc. Am. Math. Soc., 8, 303 (1957).
³ A. I. Lichtman, DAN, 152, No. 4, 812 (1963).
⁴ S. Amitsur, Trans. Am. Math. Soc., 80, 864 (1955).
⁵ I. N. Herstein, Pacific J. Math., 3, 864 (1953).
⁶ M. Sh. Khuzurbazar, DAN, 137, No. 1, 42 (1961).
⁷ M. Sh. Khuzurbazar, DAN, 131, No. 6, 1268 (1960).
⁸ A. E. Zalesskii, Matem. sborn., 67, No. 1, 154 (1965).

* Finite dimensionality of \(T\) over the field of rational numbers is not assumed.

** Note in proof. In (⁸) it is shown that all locally solvable and all periodic normal divisors of the group \(D^*\) are contained in the center.